Introduction to Riemannian Manifolds

Topological Preliminaries

An n-dimensional topological manifold (or simply an n-manifold ) is a second-countable Hausdorff topological space that is locally Euclidean of dimension ${\varvec{n}}$ , meaning that every point has a neighborhood homeomorphic to an open subset of $\mathbb {R}^n$ . Given an n -manifold M , a coordinate chart for ${\varvec{M}}$ is a pair $(U,\varphi )$ , where $U\subseteq M$ is an open set and $\varphi :U\mathrel {\rightarrow }\hat{U}$ is a homeomorphism from U to an open subset $\hat{U}\subseteq \mathbb {R}^n$ . If $p\in M$ and $(U,\varphi )$ is a chart such that $p\in U$ , we say that $(U,\varphi )$ is a chart containing ${\varvec{p}}$ .

We also occasionally need to consider manifolds with boundary. An n - dimensional topological manifold with boundary is a second-countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of $\mathbb {R}^n$ or to a (relatively) open subset of the half-space $\mathbb {R}^n_+ = \{(x^1,\dots ,x^n)\in \mathbb {R}^n: x^n\ge 0\}$ . A pair $(U,\varphi )$ , where U is an open subset of M and $\varphi$ is a homeomorphism from U to an open subset of $\mathbb {R}^n$ or $\mathbb {R}^n_+$ , is called an interior chart if $\varphi (U)$ is an open subset of $\mathbb {R}^n$ or an open subset of $\mathbb {R}^n_+$ that does not meet $\partial \mathbb {R}^n_+=\{x\in \mathbb {R}^n_+: x^n=0\}$ ; and it is called a boundary chart if $\varphi (U)$ is an open subset of $\mathbb {R}^n_+$ with ${\varphi (U)\cap \partial \mathbb {R}^n_+}\ne {\varnothing }$ . A point $p\in M$ is called an interior point of ${\varvec{M}}$ if it is in the domain of an interior chart, and it is a boundary point of ${\varvec{M}}$ if it is in the domain of a boundary chart taking p to a point of $\partial \mathbb {R}^n_+$ .

Notice that our convention is that a manifold without further qualification is always assumed to be a manifold without boundary, and the term “manifold with boundary” encompasses ordinary manifolds as a special case. But for clarity, we sometimes use redundant phrases such as “manifold without boundary” for an ordinary manifold, and “manifold with or without boundary” when we wish to emphasize that the discussion applies equally well in either case.

Proposition A.1.

[ LeeTM , Props. 4.23 & 4.64] Every topological manifold with or without boundary is locally path-connected and locally compact.

Proposition A.2.

[ LeeTM , Thm. 4.77] Every topological manifold with or without boundary is paracompact.

If M and N are topological spaces, a map $F:M\mathrel {\rightarrow }N$ is said to be a closed map if for each closed subset $K\subseteq M$ , the image set F ( K ) is closed in N , and an open map if for each open subset $U\subseteq M$ , the image set F ( U ) is open in N . It is a quotient map if it is surjective and $V\subseteq N$ is open if and only if $F^{-1}(V)$ is open.

Proposition A.3.

[ LeeTM , Prop. 3.69] Suppose M and N are topological spaces, and $F:M\mathrel {\rightarrow }N$ is a continuous map that is either open or closed.

(a)
If F is surjective, it is a quotient map.
(b)
If F is injective, it is a topological embedding.
(c)
If F is bijective, it is a homeomorphism.

The next lemma often gives a convenient shortcut for proving that a map is closed.

Lemma A.4

(Closed Map Lemma). [ LeeTM , Lemma 4.50] If F is a continuous map from a compact space to a Hausdorff space, then F is a closed map.

Here is another condition that is frequently useful for showing that a map is closed. A map $F:M\mathrel {\rightarrow }N$ between topological spaces is called a proper map if for each compact subset $K\subseteq N$ , the preimage $F^{-1}(K)$ is compact.

Proposition A.5

(Proper Continuous Maps Are Closed). [ LeeTM , Thm. 4.95] Suppose M is a topological space, N is a locally compact Hausdorff space, and $F:M\mathrel {\rightarrow }N$ is continuous and proper. Then F is a closed map.

Fundamental Groups

Many of the deepest theorems in Riemannian geometry express relations between local geometric properties and global topological properties. Because the global properties are frequently expressed in terms of fundamental groups, we summarize some basic definitions and properties here.

Suppose M and N are topological spaces. If $F_0,F_1:M\mathrel {\rightarrow }N$ are continuous maps, a homotopy from ${\varvec{F_0}}$ to ${\varvec{F_1}}$ is a continuous map $H:M \times [0, 1]\mathrel {\rightarrow }N$ that satisfies $$H(x,0)=F_0(x)$$ and $$H(x,1)=F_1(x)$$ for all $x\in M$ . If there exists such a homotopy, we say that $$F_0$$ and $$F_1$$ are homotopic .

We are most interested in homotopies in the following situation. If M is a topological space, a path in ${\varvec{M}}$ is a continuous map $\alpha :[0, 1]\mathrel {\rightarrow }M$ . If $p=\alpha (0)$ and $q=\alpha (1)$ , we say that $\alpha$ is a path from ${\varvec{p}}$ to ${\varvec{q}}$ . If $\alpha _0$ and $\alpha _1$ are both paths from p to q , a path homotopy from ${\varvec{\alpha _0}}$ to ${\varvec{\alpha _1}}$ is a continuous map $H:[0, 1]\times [0, 1]\mathrel {\rightarrow }M$ that satisfies

$\begin{aligned}&H(s,0) = \alpha _0(s) \text { and } H(s,1) = \alpha _1(s) \text { for all}~s\in [0, 1],\\&\quad H(0,t) = p \text { and } H(1,t) = q \text { for all}~t\in [0, 1]. \end{aligned}$

If there exists such a path homotopy, then $\alpha _0$ and $\alpha _1$ are said to be path-homotopic ; this is an equivalence relation on the set of all paths in M from p to q . The equivalence class of a path $\alpha$ , called its path class , is denoted by $[\alpha ]$ .

Paths $\alpha$ and $\beta$ that satisfy $\alpha (1) = \beta (0)$ are said to be composable ; in this case, their product is the path $\alpha \cdot \beta :[0, 1]\mathrel {\rightarrow }M$ defined by

$\begin{aligned} \alpha \cdot \beta (s) = \left\{ \begin{array}{ll} \alpha (2s), &{}0\le s \le \tfrac{1}{2},\\ \beta (2s-1), &{}\tfrac{1}{2} \le s \le 1. \end{array}\right. \end{aligned}$

Path products are well defined on path classes: if $[\alpha _0] = [\alpha _1]$ and $[\beta _0]=[\beta _1]$ , and $\alpha _0$ and $\beta _0$ are composable, then so are $\alpha _1$ and $\beta _1$ , and $[\alpha _0 \cdot \beta _0] = [\alpha _1 \cdot \beta _1]$ . Thus we obtain a well-defined product of path classes by $[\alpha ] \cdot [\beta ] = [\alpha \cdot \beta ]$ .

A path from a point p to itself is called a loop based at ${\varvec{p}}$ . If M is a topological space and $p\in M$ , then any two loops based at p are composable, and the set of path classes of loops based at p is a group under path class products, called the fundamental group of ${\varvec{M}}$ based at ${\varvec{p}}$ and denoted by $\pi _1(M,p)$ . The class of the constant path $c_p(s) \equiv p$ is the identity element, and the class of the reverse path $\alpha ^{-1}(s) = \alpha (1-s)$ is the inverse of $[\alpha ]$ . (Although multiplication of path classes is associative, path products themselves are not, so a multiple product such as $\alpha \cdot \beta \cdot \gamma$ is to be interpreted as $(\alpha \cdot \beta ) \cdot \gamma$ .)

Proposition A.6

(Induced Homomorphisms). [ LeeTM , Prop. 7.24 & Cor. 7.26] Suppose M and N are topological spaces and $p\in M$ . If $F:M\mathrel {\rightarrow }N$ is a continuous map, then the map $F_*:\pi _1(M,p)\mathrel {\rightarrow }\pi _1(N,F(p))$ defined by $F_*([\alpha ]) = [F\circ \alpha ]$ is a group homomorphism called the homomorphism induced by ${\varvec{F}}$ . If F is a homeomorphism, then $$F_*$$ is an isomorphism.

A topological space M is said to be simply connected if it is path-connected and for some (hence every) point $p\in M$ , the fundamental group $\pi _1(M,p)$ is trivial.

$\blacktriangleright$ Exercise A.7. Show that if M is a simply connected topological space and p , q are any two points in M , then all paths in M from p to q are path-homotopic.

A continuous map $F:M\mathrel {\rightarrow }N$ is said to be a homotopy equivalence if there is a continuous map $G:N\mathrel {\rightarrow }M$ such that $G\circ F$ and $F\circ G$ are both homotopic to identity maps.

Proposition A.8

(Homotopy Invariance of the Fundamental Group). [ LeeTM , Thm. 7.40] Suppose M and N are topological spaces and $F:M\mathrel {\rightarrow }N$ is a homotopy equivalence. Then for every point $p\in M$ , the induced homomorphism $F_*:\pi _1(M,p)\mathrel {\rightarrow }\pi _1(N,F(p))$ is an isomorphism.

Smooth Manifolds and Smooth Maps

The setting for the study of Riemannian geometry is provided by smooth manifolds , which are topological manifolds endowed with an extra structure that allows us to differentiate functions and maps. First, we note that when U is an open subset of $\mathbb {R}^n$ , a map $F:U\mathrel {\rightarrow }\mathbb {R}^k$ is said to be smooth ( or of class ${\varvec{C^\infty }}$ ) if all of its component functions have continuous partial derivatives of all orders. More generally, if the domain U is an arbitrary subset of $\mathbb {R}^n$ , not necessarily open (such as a relatively open subset of $\mathbb {R}^n_+$ ), then F is said to be smooth if for each $x\in U$ , F has a smooth extension to a neighborhood of x in $\mathbb {R}^n$ . A diffeomorphism is a bijective smooth map whose inverse is also smooth.

If M is a topological n -manifold with or without boundary, then two coordinate charts $(U,\varphi )$ , $(V,\psi )$ for M are said to be smoothly compatible if both of the transition maps $\psi \circ \varphi ^{-1}$ and $\varphi \circ \psi ^{-1}$ are smooth where they are defined (on $\varphi (U\cap V)$ and $\psi (U\cap V)$ , respectively). Since these maps are inverses of each other, it follows that both transition maps are in fact diffeomorphisms. An atlas for ${\varvec{M}}$ is a collection of coordinate charts whose domains cover M . It is called a smooth atlas if any two charts in the atlas are smoothly compatible. A smooth structure on ${\varvec{M}}$ is a smooth atlas that is maximal , meaning that it is not properly contained in any larger smooth atlas. A smooth manifold is a topological manifold endowed with a specific smooth structure; and similarly, a smooth manifold with boundary is a topological manifold with boundary endowed with a smooth structure. (We usually just say “ M is a smooth manifold,” or “ M is a smooth manifold with boundary,” with the smooth structure understood from the context.) If M is a set, a smooth manifold structure on M is a second countable, Hausdorff, locally Euclidean topology together with a smooth structure making it into a smooth manifold.

$\blacktriangleright$ Exercise A.9. Let M be a topological manifold with or without boundary. Show that every smooth atlas for M is contained in a unique maximal smooth atlas, and two smooth atlases determine the same smooth structure if and only if their union is a smooth atlas.

The manifolds in this book are always assumed to be smooth. As in most parts of differential geometry, the theory still works under weaker differentiability assumptions (such as k times continuously differentiable), but such considerations are usually relevant only when one is treating questions of hard analysis that are beyond our scope.

If M is a smooth manifold with or without boundary, then every coordinate chart in the given maximal atlas is called a smooth coordinate chart for ${\varvec{M}}$ or just a smooth chart . The set U is called a smooth coordinate domain ; if its image is an open ball in $\mathbb R^n$ , it is called a smooth coordinate ball . If in addition $\varphi$ extends to a smooth coordinate map $\varphi ':U'\mathrel {\rightarrow }\mathbb {R}^n$ on an open set $U'\supseteq \bar{U}$ such that $\varphi '\big (\bar{U}\big )$ is the closure of the open ball $\varphi (U)$ in $\mathbb {R}^n$ , then U is called a regular coordinate ball . In this case, $\bar{U}$ is diffeomorphic to a closed ball and is thus compact.

Proposition A.10.

[ LeeSM , Prop. 1.19] Every smooth manifold has a countable basis of regular coordinate balls.

Given a smooth chart $(U,\varphi )$ for M , the component functions of $\varphi$ are called local coordinates for ${\varvec{M}}$ , and are typically written as $\left( x^1,\dots ,x^n\right)$ , $\left( x^i\right)$ , or x , depending on context. Although, formally speaking, a coordinate chart is a map from an open subset $U\subseteq M$ to $\mathbb {R}^n$ , it is common when one is working in the domain of a specific chart to use the coordinate map to identify U with its image in $\mathbb {R}^n$ , and to identify a point in U with its coordinate representation $\big (x^1,\dots ,x^n\big )\in \mathbb {R}^n$ .

In this book, we always write coordinates with upper indices, as in $\big (x^i\big )$ , and expressions with indices are interpreted using the Einstein summation convention : if in some monomial term the same index name appears exactly twice, once as an upper index and once as a lower index, then that term is understood to be summed over all possible values of that index (usually from 1 to the dimension of the space). Thus the expression $$a^i v_i$$ is to be read as $\sum _{i}a^i v_i$ . As we will see below, index positions for other sorts of objects such as vectors and covectors are chosen whenever possible in such a way that summations that make mathematical sense obey the rule that each repeated index appears once up and once down in each term to be summed.

Because of the result of Exercise A.9, to define a smooth structure on a manifold, it suffices to exhibit a single smooth atlas for it. For example, $\mathbb {R}^n$ is a topological n -manifold, and it has a smooth atlas consisting of the single chart $\big (\mathbb {R}^n,\mathrm{Id}_{\mathbb {R}^n}\big )$ . More generally, if V is an n -dimensional vector space, then every basis $(b_1,\dots ,b_n)$ for V yields a linear basis isomorphism $B:\mathbb {R}^n\mathrel {\rightarrow }V$ by $B(x^1,\dots ,x^n) = x^ib_i$ , whose inverse is a global coordinate chart, and it is easy to check that all such charts are smoothly compatible. Thus every finite-dimensional vector space has a natural smooth structure, which we call its standard smooth structure . We always consider $\mathbb {R}^n$ or any other finite-dimensional vector space to be a smooth manifold with this structure unless otherwise specified.

If M is a smooth n -manifold with or without boundary and $W\subseteq M$ is an open subset, then W has a natural smooth structure consisting of all smooth charts $(U,\varphi )$ for M such that $U\subseteq W$ , so every open subset of a smooth n -manifold is a smooth n -manifold in a natural way, and similarly for manifolds with boundary. If $M_1,\dots ,M_k$ are smooth manifolds of dimensions $n_1,\dots ,n_k$ , respectively, then their Cartesian product $M_1\times \dots \times M_k$ has a natural smooth atlas consisting of charts of the form $(U_1\times \dots \times U_k, \varphi _1\times \dots \times \varphi _k)$ , where $(U_i,\varphi _i)$ is a smooth chart for $$M_i$$ ; thus the product space is a smooth manifold of dimension $n_1+\dots +n_k$ .

Suppose M and N are smooth manifolds with or without boundary. A map $F:M\mathrel {\rightarrow }N$ is said to be smooth if for every $p\in M$ , there exist smooth charts $(U,\varphi )$ for M containing p and $(V,\psi )$ for N containing F ( p ) such that $F(U)\subseteq V$ and the composite map $\psi \circ F\circ \varphi ^{-1}$ is smooth from $\varphi (U)$ to $\psi (V)$ . In particular, if N is an open subset of $\mathbb {R}^k$ or $\mathbb {R}^k_+$ with its standard smooth structure, we can take $\psi$ to be the identity map of N , and then smoothness of F just means that each point of M is contained in the domain of a chart $(U,\varphi )$ such that $F\circ \varphi ^{-1}$ is smooth. It is an easy consequence of the definition that identity maps, constant maps, and compositions of smooth maps are all smooth. A map $F:M\mathrel {\rightarrow }N$ is said to be a diffeomorphism if it is smooth and bijective and $F^{-1}:N\mathrel {\rightarrow }M$ is also smooth.

If $F:M\mathrel {\rightarrow }N$ is smooth, and $(U,\varphi )$ and $(V,\psi )$ are any smooth charts for M and N respectively, the map $\hat{F} = \psi \circ F \circ \varphi ^{-1}$ is a smooth map between appropriate subsets of $\mathbb {R}^n$ or $\mathbb {R}^n_+$ , called a coordinate representation of ${\varvec{F}}$ . Often, in keeping with the practice of using local coordinates to identify an open subset of a manifold with an open subset of $\mathbb {R}^n$ , we identify the coordinate representation of F with (the restriction of) F itself, and write things like “ F is given in local coordinates by $$F(x,y) = (xy, x-y)$$ ,” when we really mean that $\hat{F}$ is given by this formula for suitable choices of charts in the domain and codomain.

We let $C^\infty (M,N)$ denote the set of all smooth maps from M to N , and $C^\infty (M)$ the vector space of all smooth functions from M to $\mathbb {R}$ . For every function $f:M\mathrel {\rightarrow }\mathbb {R}$ or $\mathbb {R}^k$ , we define the support of ${\varvec{f}}$ , denoted by $\mathrm{supp}\,f$ , to be the closure of the set $\{x\in M: f(x)\ne 0\}$ . If $A\subseteq M$ is a closed subset and $U\subseteq M$ is an open subset containing A , then a smooth bump function for ${\varvec{A}}$ supported in ${\varvec{U}}$ is a smooth function $f:M\mathrel {\rightarrow }\mathbb {R}$ satisfying $0\le f(x)\le 1$ for all $x\in M$ , $f|_A\equiv 1$ , and $\mathrm{supp}\,f\subseteq U$ .

If ${\mathscr {U}} = \{U_\alpha \}_{\alpha \in A}$ is an indexed open cover of M , then a smooth partition of unity subordinate to ${\varvec{\mathscr {U}}}$ is an indexed family $\{\psi _\alpha \}_{\alpha \in A}$ of functions $\psi _\alpha \in C^\infty (M)$ satisfying

$0\le \psi _\alpha (x)\le 1$ for all $\alpha \in A$ and all $x\in M$ .
$\mathrm{supp}\,\, \psi _\alpha \subseteq U_\alpha$ for each $\alpha \in A$ .
The family of supports $\{\mathrm{supp}\,\, \psi _\alpha \}_{\alpha \in A}$ is locally finite : every point of M has a neighborhood that intersects $\mathrm{supp}\,\,\psi _\alpha$ for only finitely many values of $\alpha$ .
$\sum _{\alpha \in A}\psi _\alpha (x) = 1$ for all $x\in M$ .

Proposition A.11

(Existence of Partitions of Unity). [ LeeSM , Thm. 2.23] If M is a smooth manifold with or without boundary and $\mathscr {U}=\{U_\alpha \}_{\alpha \in A}$ is an indexed open cover of M , then there exists a smooth partition of unity subordinate to $\mathscr {U}$ .

Proposition A.12

(Existence of Smooth Bump Functions). [ LeeSM , Prop. 2.25] If M is a smooth manifold with or without boundary, $A\subseteq M$ is a closed subset, and $U\subseteq M$ is an open subset containing A , then there exists a smooth bump function for A supported in U .

Tangent Vectors

Let M be a smooth manifold with or without boundary. There are various equivalent ways of defining tangent vectors on M . The following definition is the most convenient to work with in practice. For every point $p\in M$ , a tangent vector at ${\varvec{p}}$ is a linear map $v:C^\infty (M)\mathrel {\rightarrow }\mathbb {R}$ that is a derivation at ${\varvec{p}}$ , meaning that for all $f,g\in C^\infty (M)$ it satisfies the product rule

$\begin{aligned} v(fg) = f(p)vg + g(p)vf. \end{aligned}$

(A.1)

The set of all tangent vectors at p is denoted by $$T_pM$$

and called the tangent space at ${\varvec{p}}$ .

Suppose M is n -dimensional and $\varphi :U\mathrel {\rightarrow }\hat{U}\subseteq \mathbb {R}^n$ is a smooth coordinate chart on some open subset $U\subseteq M$ . Writing the coordinate functions of $\varphi$ as $\big (x^1,\dots ,x^n\big )$ , we define the coordinate vectors $\partial /\partial x^1|_p$ , ..., $\partial /\partial x^n|_p$ by

$\begin{aligned} \left. \frac{\partial }{\partial x^i}\right| _p f = \left. \frac{\partial }{\partial x^i}\right| _{\varphi (p)} \bigl (f\circ \varphi ^{-1}\big ). \end{aligned}$

(A.2)

These vectors form a basis for $$T_pM$$

, which therefore has dimension n . When there can be no confusion about which coordinates are meant, we usually abbreviate $\partial /\partial x^i|_p$ by the notation $\partial _i|_p$ . Thus once a smooth coordinate chart has been chosen, every tangent vector $v\in T_pM$ can be written uniquely in the form

$\begin{aligned} v = v^i \partial _i|_p = v^1\partial _1|_p + \dots + v^n\partial _n|_p, \end{aligned}$

(A.3)

where the components $v^1,\dots ,v^n$ are obtained by applying v to the coordinate functions: $v^i = v\big (x^i\big )$ .

On a finite-dimensional vector space V with its standard smooth manifold structure, there is a natural (basis-independent) identification of each tangent space $$T_pV$$

with V itself, obtained by identifying a vector $v\in V$ with the derivation $$D_v|_p$$

, defined by

$\begin{aligned} D_v\big |_p(f) = \left. \frac{d}{dt} \right| _{t=0} f(p+tv). \end{aligned}$

In terms of the coordinates $\big (x^i\big )$ induced on V by any basis, this is the correspondence $\big (v^1,\dots ,v^n\big ) \leftrightarrow \sum _i v^i\partial _i|_p$ .

If $F:M\mathrel {\rightarrow }N$ is a smooth map and p is any point in M , we define a linear map $dF_p:T_pM\mathrel {\rightarrow }T_{F(p)}N$ , called the differential of ${\varvec{F}}$ at ${\varvec{p}}$ , by

$\begin{aligned} dF_p(v)f = v(f\circ F), \quad v\in T_pM. \end{aligned}$

$\blacktriangleright$ Exercise A.13. Given a smooth map $F:M\mathrel {\rightarrow }N$ and a point $p\in M$ , show that $$dF_p$$ is a well-defined linear map from $$T_pM$$ to $T_{F(p)}N$ . Show that the differential of the identity is the identity, and the differential of a composition is given by $d(G\circ F)_p = dG_{F(p)} \circ dF_p$ .

Once we have chosen local coordinates $\big (x^i\big )$ for M and $\big (y^j\big )$ for N , we find by unwinding the definitions that the coordinate representation of the differential is given by the Jacobian matrix of the coordinate representation of F , which is its matrix of first-order partial derivatives:

$\begin{aligned} dF_p \left( v^i \left. \frac{\partial }{\partial x^i}\right| _p\right) = \frac{\partial \hat{F}^j}{\partial x^i}(p)\, v^i \left. \frac{\partial }{\partial y^j}\right| _{F(p)}. \end{aligned}$

For every $p\in M$ , the dual vector space to $$T_pM$$

is called the cotangent space at ${\varvec{p}}$ . This is the space $$T_p^*M = (T_pM)^*$$

of real-valued linear functionals on $$T_pM$$

, called ( tangent ) covectors at ${\varvec{p}}$ . For every $f\in C^\infty (M)$ and $p\in M$ , there is a covector $df_p\in T_p^*M$ called the differential of ${\varvec{f}}$ at ${\varvec{p}}$ , defined by

$\begin{aligned} df_p(v) = vf \text { for all}~v\in T_pM. \end{aligned}$

(A.4)

Submanifolds

The theory of submanifolds is founded on the inverse function theorem and its corollaries.

Theorem A.14

(Inverse Function Theorem for Manifolds). [ LeeSM , Thm. 4.5] Suppose M and N are smooth manifolds and $F:M\rightarrow N$ is a smooth map. If the linear map $$dF_p$$ is invertible at some point $p\in M$ , then there exist connected neighborhoods $$U_0$$ of p and $$V_0$$ of F ( p ) such that $F|_{U_0}:U_0\mathrel {\rightarrow }V_0$ is a diffeomorphism.

The most useful consequence of the inverse function theorem is the following. A smooth map $F:M\mathrel {\rightarrow }N$ is said to have constant rank if the linear map $$dF_p$$ has the same rank at every point $p\in M$ .

Theorem A.15

(Rank Theorem). [ LeeSM , Thm. 4.12] Suppose M and N are smooth manifolds of dimensions m and n , respectively, and $F:M\rightarrow N$ is a smooth map with constant rank r . For each $p\in M$ there exist smooth charts $(U,\varphi )$ for M centered at p and $(V,\psi )$ for N centered at F ( p ) such that $F(U)\subseteq V,$ in which F has a coordinate representation of the form

$\begin{aligned} \hat{F}\left( x^1,\dots ,x^r,x^{r+1},\dots ,x^m\right) = \left( x^1,\dots ,x^r,0,\dots ,0\right) . \end{aligned}$

(A.5)

Here are the most important types of constant-rank maps. In all of these definitions, M and N are smooth manifolds, and $F:M\mathrel {\rightarrow }N$ is a smooth map.

F is a submersion if its differential is surjective at each point, or equivalently if it has constant rank equal to $\dim N$ .
F is an immersion if its differential is injective at each point, or equivalently if it has constant rank equal to $\dim M$ .
F is a local diffeomorphism if every point $p\in M$ has a neighborhood U such that is a diffeomorphism onto an open subset of N , or equivalently if F is both a submersion and an immersion.
F is a smooth embedding if it is an injective immersion that is also a topological embedding (a homeomorphism onto its image, endowed with the subspace topology).

Theorem A.16

(Local Embedding Theorem). [ LeeSM , Thm. 4.25] Every smooth immersion is locally an embedding: if $F:M\mathrel {\rightarrow }N$ is a smooth immersion, then for every $p\in M$ , there is a neighborhood U of p in M such that $$F|_U$$ is an embedding.

Theorem A.17

(Local Section Theorem). [ LeeSM , Thm. 4.26] Suppose M and N are smooth manifolds and $\pi :M\mathrel {\rightarrow }N$ is a smooth map. Then $\pi$ is a smooth submersion if and only if every point of M is in the image of a smooth local section of ${\varvec{\pi }}$ (a map $\sigma :U\mathrel {\rightarrow }M$ defined on some open subset $U\subseteq N$ , with $\pi \circ \sigma = \mathrm{Id}_U$ ).

Proposition A.18

(Properties of Submersions). [ LeeSM , Prop. 4.28] Let M and N be smooth manifolds and let $\pi :M\mathrel {\rightarrow }N$ be a smooth submersion.

(a)
$\pi$ is an open map.
(b)
If $\pi$ is surjective, it is a quotient map.

Proposition A.19

(Uniqueness of Smooth Quotients). [ LeeSM , Thm. 4.31] Suppose M , $$N_1$$ , and $$N_2$$ are smooth manifolds, and $\pi _1:M\mathrel {\rightarrow }N_1$ and $\pi _2:M\mathrel {\rightarrow }N_2$ are surjective smooth submersions that are constant on each other’s fibers. Then there exists a unique diffeomorphism $F:N_1\mathrel {\rightarrow }N_2$ such that $F\circ \pi _1 = \pi _2$ .

Theorem A.20

(Global Rank Theorem). [ LeeSM , Thm. 4.14] Suppose M and N are smooth manifolds, and $F:M\mathrel {\rightarrow }N$ is a smooth map of constant rank.

(a)
If F is surjective, then it is a smooth submersion.
(b)
If F is injective, then it is a smooth immersion.
(c)
If F is bijective, then it is a diffeomorphism.

Suppose ${{\widetilde{M}}}$ is a smooth manifold with or without boundary. An immersed n-dimensional submanifold of ${\varvec{{{\widetilde{M}}}}}$ is a subset $M\subseteq {{\widetilde{M}}}$ endowed with a topology that makes it into an n -dimensional topological manifold and a smooth structure such that the inclusion map $M\mathrel {\hookrightarrow }{{\widetilde{M}}}$ is a smooth immersion. It is called an embedded submanifold if the inclusion is a smooth embedding, or equivalently if the given topology on M is the subspace topology. Immersed and embedded submanifolds with boundary are defined similarly. In each case, the codimension of ${\varvec{M}}$ is the difference $\dim {{\widetilde{M}}}- \dim M$ . A submanifold of codimension 1 is known as a hypersurface . Without further qualification, the word submanifold means an immersed submanifold, which includes an embedded one as a special case.

An embedded submanifold with or without boundary is said to be properly embedded if the inclusion map is proper.

Proposition A.21.

[ LeeSM , Prop. 5.5] If ${{\widetilde{M}}}$ is a smooth manifold with or without boundary, then an embedded submanifold $M\subseteq {{\widetilde{M}}}$ is properly embedded if and only if it is a closed subset of ${{\widetilde{M}}}$ .

A properly embedded codimension-0 submanifold with boundary in ${{\widetilde{M}}}$ is called a regular domain .

Proposition A.22

(Slice Coordinates). [ LeeSM , Thms. 5.8 & 5.51] Let ${{\widetilde{M}}}$ be a smooth m -manifold without boundary and let $M\subseteq {{\widetilde{M}}}$ be an embedded n -dimensional submanifold with or without boundary. Then for each $p\in M$ there exist a neighborhood ${\smash [t]{\tilde{U}}}$ of p in ${{\widetilde{M}}}$ and smooth coordinates $\big (x^1,\dots ,x^m\big )$ for ${{\widetilde{M}}}$ on ${\smash [t]{\tilde{U}}}$ such that $M \cap {\smash [t]{\tilde{U}}}$ is a set of one of the following forms:

$\begin{aligned} M\cap {\smash [t]{\tilde{U}}}= \left\{ \begin{array}{l@{\quad }l} \{x\in {\smash [t]{\tilde{U}}}:x^{n+1}=\dots =x^m=0\} &{}\text {if}~p\notin \partial M,\\ \{x\in {\smash [t]{\tilde{U}}}:x^{n+1}=\dots =x^m=0 \text { and }x^n\ge 0\}&{}\text {if}~p\in \partial M.\end{array}\right. \end{aligned}$

In such a chart, $\left( x^1,\dots ,x^n\right)$ form smooth local coordinates for M .

Coordinates satisfying either of these conditions are called slice coordinates ; when it is necessary to distinguish them, coordinates satisfying the second condition are referred to as boundary slice coordinates .

If M is an immersed submanifold, then Theorem A.16 shows that every point of M has a neighborhood U in M that is embedded in ${{\widetilde{M}}}$ , so we can construct slice coordinates for U (but they need not be slice coordinates for M ).

$\blacktriangleright$ Exercise A.23. Suppose ${{\widetilde{M}}}$ is a smooth manifold and $M\subseteq {{\widetilde{M}}}$ is an embedded submanifold. Show that every smooth function $f: M \mathrel {\rightarrow }{\mathbb {R}}$ can be extended to a smooth function on a neighborhood of M in ${{\widetilde{M}}}$ whose restriction to M is f ; and if M is also closed in ${{\widetilde{M}}}$ , then the neighborhood can be taken to be all of ${{\widetilde{M}}}$ . [Hint: Extend f locally in slice coordinates by letting it be independent of $\big (x^{n+1},\dots ,x^m\big )$ , and patch together using a partition of unity.]

Here is the way that most submanifolds are presented. Suppose $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }N$ is any map. Every subset of the form $\varPhi ^{-1}(\{y\})\subseteq {{\widetilde{M}}}$ for some $y\in N$ is called a level set of ${\varvec{\varPhi }}$ , or the fiber of ${\varvec{\varPhi }}$ over ${\varvec{y}}$ . The simpler notation $\varPhi ^{-1}(y)$ is also used for a level set when there is no likelihood of ambiguity.

Theorem A.24

(Constant-Rank Level Set Theorem). [ LeeSM , Thm. 5.12] Suppose ${{\widetilde{M}}}$ and N are smooth manifolds, and $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }N$ is a smooth map with constant rank r . Every level set of $\varPhi$ is a properly embedded submanifold of codimension r in ${{\widetilde{M}}}$ .

Corollary A.25

(Submersion Level Set Theorem). [ LeeSM , Cor. 5.13] Suppose ${{\widetilde{M}}}$ and N are smooth manifolds, and $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }N$ is a smooth submersion. Every level set of $\varPhi$ is a properly embedded submanifold of ${{\widetilde{M}}}$ , whose codimension is equal to $\dim N$ .

In fact, a map does not have to be a submersion, or even to have constant rank, for its level sets to be embedded submanifolds. If $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }N$ is a smooth map, a point $p\in {{\widetilde{M}}}$ is called a regular point of ${\varvec{\varPhi }}$ if the linear map $d\varPhi _p:T_p{{\widetilde{M}}}\mathrel {\rightarrow }T_{\varPhi (p)}N$ is surjective, and p is called a critical point of ${\varvec{\varPhi }}$ if it is not. A point $c\in N$ is called a regular value of ${\varvec{\varPhi }}$ if every point of $\varPhi ^{-1}(c)$ is a regular point of $\varPhi$ , and a critical value otherwise. A level set $\varPhi ^{-1}(c)$ is called a regular level set of ${\varvec{\varPhi }}$ if c is a regular value of $\varPhi$ .

Corollary A.26

(Regular Level Set Theorem). [ LeeSM , Cor. 5.14] Let ${{\widetilde{M}}}$ and N be smooth manifolds, and let $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }N$ be a smooth map. Every regular level set of $\varPhi$ is a properly embedded submanifold of ${{\widetilde{M}}}$ whose codimension is equal to $\dim N$ .

Suppose ${{\widetilde{M}}}$ is a smooth manifold and $M\subseteq {{\widetilde{M}}}$ is an embedded codimension- k submanifold. A smooth map $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }\mathbb {R}^k$ is called a defining function for ${\varvec{M}}$ if M is a regular level set of $\varPhi$ . More generally, if $\varPhi :U\mathrel {\rightarrow }\mathbb {R}^k$ is a smooth map defined on an open subset $U\subseteq {{\widetilde{M}}}$ such that $M\cap U$ is a regular level set of $\varPhi$ , then $\varPhi$ is called a local defining function for ${\varvec{M}}$ . The following is a partial converse to the submersion level set theorem.

Proposition A.27

(Existence of Local Defining Functions). [ LeeSM , Prop. 5.16] Suppose ${{\widetilde{M}}}$ is a smooth manifold and $M\subseteq {{\widetilde{M}}}$ is an embedded submanifold of codimension k . For each point $p\in M$ , there exist a neighborhood U of p in ${{\widetilde{M}}}$ and a smooth submersion $\varPhi :U\mathrel {\rightarrow }\mathbb {R}^k$ such that $M\cap U$ is a level set of $\varPhi$ .

If $M\subseteq {{\widetilde{M}}}$ is a submanifold (immersed or embedded), the inclusion map $\iota :M\mathrel {\hookrightarrow }{{\widetilde{M}}}$ induces an injective linear map $d\iota _p:T_pM \mathrel {\rightarrow }T_p{{\widetilde{M}}}$ for each $p\in M$ . It is standard practice to identify $$T_pM$$ with its image $d\iota _p(T_pM)\subseteq T_p{{\widetilde{M}}}$ , thus regarding as a subspace of $T_p{{\widetilde{M}}}$ . The next proposition gives two handy ways to identify this subspace in the embedded case.

Proposition A.28

(Tangent Space to a Submanifold). [ LeeSM , Props. 5.37 & 5.38] Suppose ${{\widetilde{M}}}$ is a smooth manifold, $M\subseteq {{\widetilde{M}}}$ is an embedded submanifold, and $p\in M$ . As a subspace of $T_p{{\widetilde{M}}}$ , the tangent space $$T_pM$$

is characterized by

$\begin{aligned} T_pM = \left\{ v\in T_p{{\widetilde{M}}}: vf = 0 \text { whenever}~f\in C^\infty \big ({{\widetilde{M}}}\big )~\text {and}~{f\bigr |}_M=0\right\} . \end{aligned}$

If $\varPhi$ is a local defining function for M on a neighborhood of p , then

$\begin{aligned} T_pM = {{\,\mathrm{Ker}\,}}d\varPhi _p. \end{aligned}$

Proposition A.29

(Lagrange Multiplier Rule). Let ${{\widetilde{M}}}$ be a smooth m -manifold and $M\subseteq {{\widetilde{M}}}$ be an embedded hypersurface. Suppose $f\in C^\infty \big ({{\widetilde{M}}}\big )$ , and $p\in M$ is a point at which f attains a local maximum or minimum value among points in M . If $\varPhi :U\mathrel {\rightarrow }\mathbb {R}$ is a local defining function for M on a neighborhood U of p , then there is a real number $\lambda$ (called a Lagrange multiplier ) such that $df_p = \lambda d\varPhi _p$ .

$\blacktriangleright$ Exercise A.30. Prove the preceding proposition. [Hint: One way to proceed is to begin by showing that there are smooth functions $x^1,\dots ,x^{m-1}$ on a neighborhood of p such that $(x^1,\dots ,x^{m-1},\varPhi )$ are local slice coordinates for M .]

In treating manifolds with boundary, many constructions can be simplified by viewing a manifold with boundary as being embedded in a larger manifold without boundary. Here is one way to do that.

If M is a topological manifold with nonempty boundary, the double of ${\varvec{M}}$ is the quotient space D ( M ) formed from the disjoint union of two copies of M (say $$M_1$$ and $$M_2$$ ), by identifying each point in $\partial M_1$ with the corresponding point in $\partial M_2$ .

Proposition A.31.

[ LeeSM , Example 9.32] Suppose M is a smooth manifold with nonempty boundary. Then D ( M ) is a topological manifold without boundary, and it has a smooth structure such that the natural maps $M\mathrel {\rightarrow }M_1\mathrel {\rightarrow }D(M)$ and $M\mathrel {\rightarrow }M_2\mathrel {\rightarrow }D(M)$ are smooth embeddings. Moreover, D ( M ) is compact if and only if M is compact, and connected if and only if M is connected.

Vector Bundles

Suppose M is a smooth manifold with or without boundary. The tangent bundle of ${\varvec{M}}$ , denoted by TM , is the disjoint union of the tangent spaces at all points of M : $TM = \coprod _{p\in M} T_pM$ . This set can be thought of both as a union of vector spaces and as a manifold in its own right. This kind of structure, called a vector bundle , is extremely common in differential geometry, so we take some time to review the definitions and basic properties of vector bundles.

For any positive integer k , a smooth vector bundle of rank ${\varvec{k}}$ is a pair of smooth manifolds E and M with or without boundary, together with a smooth surjective map $\pi :E\mathrel {\rightarrow }M$ satisfying the following conditions:

(i)
For each $p\in M$ , the set $E_p= \pi ^{-1}(p)$ is endowed with the structure of a k -dimensional real vector space.
(ii)
For each $p\in M$ , there exist a neighborhood U of p and a diffeomorphism $\varPhi :\pi ^{-1}(U) \mathrel {\rightarrow }U\times \mathbb {R}^k$ such that $\pi _U\circ \varPhi = \pi$ , where $\pi _U:U\times \mathbb {R}^k\mathrel {\rightarrow }U$ is the projection onto the first factor; and for each $q\in U$ , $\varPhi$ restricts to a linear isomorphism from to ${\{q\}\times \mathbb {R}^k}\cong \mathbb {R}^k$ .

The space M is called the base of the bundle, E is called its total space , and $\pi$ is its projection . Each set $E_p = \pi ^{-1}(p)$ is called the fiber of ${\varvec{E}}$ over ${\varvec{p}}$ , and each diffeomorphism $\varPhi :\pi ^{-1}(U) \mathrel {\rightarrow }U\times \mathbb {R}^k$ described above is called a smooth local trivialization .

It frequently happens that we are presented with a collection of vector spaces (such as tangent spaces), one for each point in a manifold, that we would like to “glue together” to form a vector bundle. There is a shortcut for showing that such a collection forms a vector bundle without first constructing a smooth manifold structure on the total space. As the next lemma shows, all we need to do is to exhibit the maps that we wish to regard as local trivializations and check that they overlap correctly.

Lemma A.32

(Vector Bundle Chart Lemma). [ LeeSM , Lemma 10.6] Let M be a smooth manifold with or without boundary, and suppose that for each $p\in M$ we are given a real vector space $$E_p$$

of some fixed dimension k . Let $E=\coprod _{p\in M} E_p$ (the disjoint union of all the spaces $$E_p$$

), and let $\pi :E\mathrel {\rightarrow }M$ be the map that takes each element of $$E_p$$

to the point p . Suppose furthermore that we are given

(i)
an indexed open cover $\{U_\alpha \}_{\alpha \in A}$ of M ;
(ii)
for each $\alpha \in A$ , a bijective map $\varPhi _\alpha :\pi ^{-1}(U_\alpha )\mathrel {\rightarrow }U_\alpha \times \mathbb {R}^k$ whose restriction to each is a linear isomorphism from to $\{p\}\times \mathbb {R}^k\cong \mathbb {R}^k$ ;
(iii)
for each $\alpha ,\beta \in A$ such that $U_\alpha \cap U_\beta \ne {\varnothing }$ , a smooth map $\tau _{\alpha \beta }:U_\alpha \cap U_\beta \mathrel {\rightarrow }{{\,\mathrm{GL}\,}}(k,\mathbb {R})$ such that the composite map $\varPhi _\alpha \circ \varPhi _\beta ^{-1}$ from $(U_\alpha \cap U_\beta )\times \mathbb {R}^k$ to itself has the form
$\begin{aligned} \varPhi _\alpha \circ \varPhi _\beta ^{-1}(p,v) = (p,\tau _{\alpha \beta }(p)v). \end{aligned}$

(A.6)

Then E has a unique smooth manifold structure making it a smooth vector bundle of rank k over M , with $\pi$ as projection and the maps $\varPhi _\alpha$ as smooth local trivializations.

The smooth $GL(k,\mathbb {R})$ -valued maps $\tau _{\alpha \beta }$ of this lemma are called transition functions for ${\varvec{E}}$ .

If $\pi :E\mathrel {\rightarrow }M$ is a smooth vector bundle over M , then a section of ${\varvec{E}}$ is a continuous map $\sigma :M\mathrel {\rightarrow }E$ such that $\pi \circ \sigma = \mathrm{Id}_M$ , or equivalently, $\sigma (p)\in E_p$ for all p . If $U\subseteq M$ is an open set, then a local section of ${\varvec{E}}$ over ${\varvec{U}}$ is a continuous map $\sigma :U\mathrel {\rightarrow }E$ satisfying the analogous equation $\pi \circ \sigma = \mathrm{Id}_U$ . A smooth ( local or global ) section of ${\varvec{E}}$ is just a section that is smooth as a map between smooth manifolds. If we need to speak of “sections” that are not necessarily continuous, we use the following terminology: a rough ( local ) section of ${\varvec{E}}$ is a map $\sigma :U\mathrel {\rightarrow }E$ defined on some open set $U\subseteq M$ , not necessarily smooth or even continuous, such that $\pi \circ \sigma = \mathrm{Id}_U$ .

A local frame for E is an ordered k -tuple $(\sigma _1,\dots ,\sigma _k)$ of local sections over an open set U whose values at each $p\in U$ constitute a basis for $$E_p$$

. It is called a global frame if $$U=M$$

. If $\tau$ is a (local or global) section of E and $(\sigma _1,\dots ,\sigma _k)$ is a local frame for E over $U\subseteq M$ , then the value of $\tau$ at each $p\in U$ can be written

$\begin{aligned} \tau (p) = \tau ^i(p) \sigma _i(p) \end{aligned}$

for some functions $\tau ^1,\dots ,\tau ^k:U\mathrel {\rightarrow }\mathbb {R}$ , called the component functions of $\varvec{\sigma }$ with respect to the given frame.

The next lemma gives a criterion for smoothness that is usually easy to verify in practice.

Lemma A.33

(Local Frame Criterion for Smoothness). [ LeeSM , Prop. 10.22] Let $\pi :E\mathrel {\rightarrow }M$ be a smooth vector bundle, and let $\tau :M\mathrel {\rightarrow }E$ be a rough section. If $(\sigma _i)$ is a smooth local frame for E over an open subset $U\subseteq M$ , then $\tau$ is smooth on U if and only if its component functions with respect to $(\sigma _i)$ are smooth.

If $E\mathrel {\rightarrow }M$ is a smooth vector bundle, then the set of smooth sections of E , denoted by $\varGamma (E)$ , is a vector space (usually infinite-dimensional) under pointwise addition and multiplication by constants. Its zero element is the zero section $\zeta$ defined by $\zeta (p) = 0\in E_p$ for all $p\in M$ . For every section $\sigma$ of E , the support of ${\varvec{\sigma }}$ is the closure of the set $\{p\in M: \sigma (p)\ne 0\}$ .

Given a smooth vector bundle $\pi _E:E\mathrel {\rightarrow }M$ , a smooth subbundle of ${\varvec{E}}$ is a smooth vector bundle $\pi _D:D\mathrel {\rightarrow }M$ , in which D is an embedded submanifold (with or without boundary) of E and $\pi _D=\pi _E\big |_D$ , such that for each $p \in M$ , the subset $D_p = D \cap E_p$ is a linear subspace of $$E_p$$ , and the vector space structure on $$D_p$$ is the one inherited from $$E_p$$ . Given a collection of subspaces $D_p\subseteq E_p$ , one for each $p\in M$ , the following lemma gives a convenient condition for checking that their union is a smooth subbundle.

Lemma A.34

(Local Frame Criterion for Subbundles). [ LeeSM , Lemma 10.32] Let $\pi :E\mathrel {\rightarrow }M$ be a smooth vector bundle, and suppose that for each $p\in M$ we are given a k -dimensional linear subspace $D_p\subseteq E_p$ . Suppose further that each $p\in M$ has a neighborhood U on which there are smooth local sections $\sigma _1,\dots ,\sigma _k:U\mathrel {\rightarrow }E$ such that $\sigma _1(q),\dots ,\sigma _k(q)$ form a basis for $$D_q$$ at each $q\in U$ . Then $D = \bigcup _{p\in M}D_p\subseteq E$ is a smooth subbundle of E .

Suppose ${{\widetilde{M}}}$ is a smooth manifold with or without boundary, and $M\subseteq {{\widetilde{M}}}$ is an immersed or embedded submanifold with or without boundary. If $E\mathrel {\rightarrow }{{\widetilde{M}}}$ is any smooth rank- k vector bundle over ${{\widetilde{M}}}$ , we obtain a smooth vector bundle $\pi |_M:E|_M\mathrel {\rightarrow }M$ of rank k over M , called the restriction of ${\varvec{E}}$ to ${\varvec{M}}$ , whose total space is $E|_M = \pi ^{-1}(M)$ , whose fiber at each $p\in M$ is exactly the fiber of E , and whose local trivializations are the restrictions of the local trivializations of E . (See [ LeeSM , Example 10.8].)

Every smooth section of E restricts to a smooth section of $E|_{M}$ . Conversely, the next exercise shows that in most cases, smooth sections of $$E|_M$$ extend to smooth sections of E , at least locally near M .

$\blacktriangleright$ Exercise A.35. Suppose ${{\widetilde{M}}}$ is a smooth manifold, $E\mathrel {\rightarrow }{{\widetilde{M}}}$ is a smooth vector bundle, and $M\subseteq {{\widetilde{M}}}$ is an embedded submanifold. Show that every smooth section of the restricted bundle $$E|_M$$ can be extended to a smooth section of E on a neighborhood of M in ${{\widetilde{M}}}$ ; and if M is closed in ${{\widetilde{M}}}$ , the neighborhood can be taken to be all of ${{\widetilde{M}}}$ .

The Tangent Bundle and Vector Fields

We continue to assume that M is a smooth manifold with or without boundary. In this section, we see how the theory of vector bundles applies to the particular case of the tangent bundle.

Given any smooth local coordinate chart $\big (U,\big (x^i\big )\big )$ for M , we can define a local trivialization of TM over U by letting $\varPhi :\pi ^{-1}(U)\mathrel {\rightarrow }U\times \mathbb {R}^n$ be the map sending $v\in T_pM$ to $\big (p,(v^1,\dots ,v^n)\big )$ , where $v^i\partial /\partial x^i|_p$ is the coordinate representation for v . Given another such chart $\big (\tilde{U},\big (\tilde{x}^i\big )\big )$ for M , we obtain a corresponding local trivialization $\tilde{\varPhi }(v) = \big (p,(\tilde{v}^1,\dots ,\tilde{v}^n)\big )$ , where $v = \tilde{v}^i\partial /\partial \tilde{x}^i|_p$ . The transformation law for coordinate vectors shows that

$\begin{aligned} \tilde{v}^j = \frac{\partial \tilde{x}^j}{\partial x^i}(p) v^i, \end{aligned}$

so the transition function between two such charts is given by the smooth matrix-valued function $\big (\partial \tilde{x}^j/\partial x^i\big )$ . Thus the chart lemma shows that these local trivializations turn TM into a smooth vector bundle.

A smooth coordinate chart $\big (x^i\big )$ for M also yields smooth local coordinates $\big (x^1,\dots ,x^n,v^1,\dots ,v^n\big )$ on $\pi ^{-1}(U)\subseteq TM$ , called natural coordinates for ${\varvec{TM}}$ , by following the local trivialization $\varPhi$ with the map $U\times \mathbb {R}^n\mathrel {\rightarrow }\mathbb {R}^n\times \mathbb {R}^n$ that sends $\big (p,\left( v^1,\dots ,v^n\right) \big )$ to $\big (x^1(p),\dots ,x^n(p),v^1,\dots ,v^n\big )$ .

A section of TM is called a vector field on ${\varvec{M}}$ . To avoid confusion between the point $p\in M$ at which a vector field is evaluated and the action of the vector field on a function, we usually write the value of a vector field X at $p\in M$ as $X_p\in T_pM$ , or, if it is clearer (for example if X itself has one or more subscripts), as $$X|_p$$ .

For example, if $\big (U,\big (x^i\big )\big )$ is a smooth coordinate chart for M , for each i we get a smooth coordinate vector field on U , denoted by $\partial /\partial x^i$ (and often abbreviated by $\partial _i$ if no confusion results), whose value at $p\in U$ is the coordinate vector $\partial /\partial x^i|_p$ defined above. Each coordinate vector field is smooth because its coordinate representation in natural coordinates is $(x^1,\dots ,x^n) \mapsto \big (x^1,\dots ,x^n,0,\dots ,0,1,0,\dots ,0\big )$ . The vector fields $\partial _1,\dots ,\partial _n$ form a smooth local frame for TM , called a coordinate frame .

It follows from Lemma A.33 that a vector field is smooth if and only if its components are smooth with respect to some smooth local frame (such as a coordinate frame) in a neighborhood of each point. It is standard to use the notation $\mathfrak {X}(M)$ as another name for $\varGamma (TM)$ , the space of all smooth vector fields on M .

Now suppose M and N are smooth manifolds with or without boundary, and $F:M\mathrel {\rightarrow }N$ is a smooth map. We obtain a smooth map $dF:TM\mathrel {\rightarrow }TN$ , called the global differential of ${\varvec{F}}$ , whose restriction to each tangent space $$T_pM$$ is the linear map $$dF_p$$ defined above. In general, the global differential does not take vector fields to vector fields. In the special case that $X\in \mathfrak X(M)$ and $Y\in \mathfrak X(N)$ are vector fields such that $dF(X_p) = Y_{F(p)}$ for all $p\in M$ , we say that the vector fields X and Y are F-related .

Lemma A.36.

[ LeeSM , Prop. 8.19 & Cor. 8.21] Let $F:M\mathrel {\rightarrow }N$ be a diffeomorphism between smooth manifolds with or without boundary. For every $X\in \mathfrak X(M)$ , there is a unique vector field $F_*X\in \mathfrak X(N)$ , called the pushforward of ${\varvec{X}}$ , that is F -related to X . For every $f\in C^\infty (N)$ , it satisfies

$\begin{aligned} \bigl ((F_*X)f\bigr )\circ F = X(f\circ F). \end{aligned}$

(A.7)

Suppose $X\in \mathfrak X(M)$ . Given a real-valued function $f\in C^\infty (M)$ , applying X to f yields a new function $Xf\in C^\infty (M)$ by $$Xf(p) = X_pf$$

. The defining equation ( A.1 ) for tangent vectors translates into the following product rule for vector fields:

$\begin{aligned} X(fg) = f\, Xg + g\, Xf. \end{aligned}$

(A.8)

A map $X:C^\infty (M)\mathrel {\rightarrow }C^\infty (M)$ is called a derivation of ${\varvec{C^\infty (M)}}$ (as opposed to a derivation at a point ) if it is linear over $\mathbb {R}$ and satisfies ( A.8 ) for all $f,g\in C^\infty (M)$ .

Lemma A.37.

[ LeeSM , Prop. 8.15] Let M be a smooth manifold with or without boundary. A map $D:C^\infty (M)\mathrel {\rightarrow }C^\infty (M)$ is a derivation if and only if it is of the form $$Df=Xf$$ for some $X\in \mathfrak X(M)$ .

Given smooth vector fields $X,Y\in \mathfrak X(M)$ , define a map $[X,Y]:C^\infty (M)\mathrel {\rightarrow }C^\infty (M)$ by

$\begin{aligned}{}[X,Y]f = X(Yf) - Y(Xf). \end{aligned}$

A straightforward computation shows that [ X , Y ] is a derivation of $C^\infty (M)$ , and thus by Lemma A.37 it defines a smooth vector field, called the Lie bracket of ${\varvec{X}}$ and ${\varvec{Y}}$ .

Proposition A.38

(Properties of Lie Brackets). [ LeeSM , Prop. 8.28] Let M be a smooth manifold with or without boundary and $X,Y,Z\in \mathfrak X(M)$ .

(a)
[ X , Y ] is bilinear over $\mathbb {R}$ as a function of X and Y .
(b)
(antisymmetry).
(c)
(Jacobi identity).
(d)
For $f,g\in C^\infty (M)$ , .

Proposition A.39

(Naturality of Lie Brackets). [ LeeSM , Prop. 8.30 & Cor. 8.31] Let $F:M\mathrel {\rightarrow }N$ be a smooth map between manifolds with or without boundary, and let $X_1,X_2\in \mathfrak X(M)$ and $Y_1,Y_2\in \mathfrak X(N)$ be vector fields such that $$X_i$$ is F -related to $$Y_i$$ for $$i=1,2$$ . Then $$[X_1,X_2]$$ is F -related to $$[Y_1,Y_2]$$ . In particular, if F is a diffeomorphism, then $$F_*[X_1,X_2] = [F_*X_1,F_*X_2]$$ .

Now suppose ${{\widetilde{M}}}$ is a smooth manifold with or without boundary and $M\subseteq {{\widetilde{M}}}$ is an immersed or embedded submanifold with or without boundary. The bundle $T{{\widetilde{M}}}|_{M}$ , obtained by restricting $T{{\widetilde{M}}}$ to M , is called the ambient tangent bundle . It is a smooth bundle over M whose rank is equal to the dimension of ${{\widetilde{M}}}$ . The tangent bundle TM is naturally viewed as a smooth subbundle of $T{{\widetilde{M}}}|_M$ , and smooth vector fields on M can also be viewed as smooth sections of $T{{\widetilde{M}}}|_M$ . A vector field $X\in \mathfrak X\big ({{\widetilde{M}}}\big )$ always restricts to a smooth section of $T{{\widetilde{M}}}|_M$ , and it restricts to a smooth section of TM if and only if it is tangent to ${\varvec{M}}$ , meaning that $X_p\in T_pM\subseteq T_p{{\widetilde{M}}}$ for each $p\in M$ .

Corollary A.40

(Brackets of Vector Fields Tangent to Submanifolds). [ LeeSM , Cor. 8.32] Let ${{\widetilde{M}}}$ be a smooth manifold and let M be an immersed submanifold with or without boundary in ${{\widetilde{M}}}$ . If $$Y_1$$ and $$Y_2$$ are smooth vector fields on ${{\widetilde{M}}}$ that are tangent to M , then $$[Y_1,Y_2]$$ is also tangent to M .

$\blacktriangleright$ Exercise A.41. Let ${{\widetilde{M}}}$ be a smooth manifold with or without boundary and let $M\subseteq {{\widetilde{M}}}$ be an embedded submanifold with or without boundary. Show that a vector field $X\in \mathfrak X\big ({{\widetilde{M}}}\big )$ is tangent to M if and only if $\big (X f\big )|_M=0$ whenever $f\in C^\infty \big ({{\widetilde{M}}}\big )$ is a function that vanishes on M .

Integral Curves and Flows

A curve in a smooth manifold M (with or without boundary) is a continuous map $\gamma :I\mathrel {\rightarrow }M$ , where $I\subseteq \mathbb {R}$ is some interval. If $\gamma$ is smooth, then for each $t_0\in I$ we obtain a vector $\gamma '(t_0) = d\gamma _{t_0}\big (d/dt|_{t_0}\big )$ , called the velocity of ${\varvec{\gamma }}$ at time ${\varvec{t_0}}$ . It acts on functions by

$\begin{aligned} \gamma '(t_0)f = (f\circ \gamma )'(t_0). \end{aligned}$

In any smooth local coordinates, the coordinate expression for $\gamma '(t_0)$ is exactly the same as it would be in $\mathbb {R}^n$ : the components of $\gamma '(t_0)$ are the ordinary t -derivatives of the components of $\gamma$ .

If $X\in \mathfrak X(M)$ , then a smooth curve $\gamma :I\mathrel {\rightarrow }M$ is called an integral curve of ${\varvec{X}}$ if its velocity at each point is equal to the value of X there: $\gamma '(t) = X_{\gamma (t)}$ for each $t\in I$ .

The fundamental fact about vector fields (at least in the case of manifolds without boundary) is that there exists a unique maximal integral curve starting at each point, varying smoothly as the point varies. These integral curves are all encoded into a global object called a flow , which we now define.

Given a smooth manifold M (without boundary), a flow domain for M is an open subset $\mathscr {D}\subseteq \mathbb {R}\times M$ with the property that for each $p\in M$ , the set $\mathscr {D}^{(p)} = {\{t\in \mathbb {R}: (t,p)\in \mathscr {D}\}}$ is an open interval containing 0. Given a flow domain $\mathscr {D}$ and a map $\theta :\mathscr {D}\mathrel {\rightarrow }M$ , for each $t\in \mathbb {R}$ we let $M_t = \{p\in M: (t,p)\in \mathscr {D}\}$ , and we define maps $\theta _t:M_t\mathrel {\rightarrow }M$ and $\theta ^{(p)}:\mathscr {D}^{(p)}\mathrel {\rightarrow }M$ by $\theta _t(p) = \theta ^{(p)}(t) = \theta (t,p)$ .

A flow on M is a continuous map $\theta :\mathscr {D}\mathrel {\rightarrow }M$ , where $\mathscr {D}\subseteq \mathbb {R}\times M$ is a flow domain, that satisfies

$\begin{aligned} \theta _0&= \mathrm{Id}_M,\\ \theta _t\circ \theta _s(p)&= \theta _{t+s}(p) \quad \text {wherever both sides are defined.} \end{aligned}$

If $\theta$ is a smooth flow, we obtain a smooth vector field $X\in \mathfrak X(M)$ defined by $X_p = \big (\theta ^{(p)}\big )'(0)$ , called the infinitesimal generator of ${\varvec{\theta }}$ .

Theorem A.42

(Fundamental Theorem on Flows). [ LeeSM , Thm. 9.12] Let X be a smooth vector field on a smooth manifold M (without boundary). There is a unique smooth maximal flow $\theta :\mathscr {D}\mathrel {\rightarrow }M$ whose infinitesimal generator is X . This flow has the following properties:

(a)
For each $p\in M,$ the curve $\theta ^{(p)}:\mathscr {D}^{(p)}\mathrel {\rightarrow }M$ is the unique maximal integral curve of X starting at p .
(b)
If $s\in \mathscr {D}^{(p)}$ , then $\mathscr {D}^{(\theta (s,p))}$ is the interval $\mathscr {D}^{(p)}-s= \bigl \{t-s: t\in \mathscr {D}^{(p)}\bigr \}$ .
(c)
For each $t\in \mathbb {R}$ , the set is open in M , and $\theta _t:M_t\mathrel {\rightarrow }M_{-t}$ is a diffeomorphism with inverse $\theta _{-t}$ .

Although the fundamental theorem guarantees only that each point lies on an integral curve that exists for a short time, the next lemma can often be used to prove that a particular integral curve exists for all time.

Lemma A.43

(Escape Lemma). Suppose M is a smooth manifold and $X\in \mathfrak X(M)$ . If $\gamma :I\mathrel {\rightarrow }M$ is a maximal integral curve of X whose domain I has a finite least upper bound b , then for every $t_0\in I$ , $\gamma \big ([t_0,b)\big )$ is not contained in any compact subset of M .

$\blacktriangleright$ Exercise A.44. Prove the escape lemma.

Proposition A.45

(Canonical Form for a Vector Field). [ LeeSM , Thm. 9.22] Let X be a smooth vector field on a smooth manifold M , and suppose $p\in M$ is a point where $X_p\ne 0$ . There exist smooth coordinates $\big (x^i\big )$ on some neighborhood of p in which X has the coordinate representation $\partial /\partial x^1$ .

The fundamental theorem on flows leads to a way of differentiating one vector field along the flow of another. Suppose M is a smooth manifold, $X,Y\in \mathfrak X(M)$ , and $\theta$ is the flow of X . The Lie derivative of ${\varvec{Y}}$ with respect to ${\varvec{X}}$ is the vector field $\mathscr {L}_X Y$ defined by

$\begin{aligned} (\mathscr {L}_X Y)_p = \left. \frac{d}{dt}\right| _{t=0} d(\theta _{-t})_{\theta _t(p)}\big (Y_{\theta _t(p)}\big ) =\lim _{t\mathrel {\rightarrow }0} \frac{ d(\theta _{-t})_{\theta _t(p)} \big ( Y_{\theta _t(p)}\big ) - Y_p}{t}. \end{aligned}$

This formula is useless for computation, however, because typically the flow of a vector field is difficult or impossible to compute explicitly. Fortunately, there is another expression for the Lie derivative that is much easier to compute.

Proposition A.46.

[ LeeSM , Thm. 9.38] Suppose M is a smooth manifold and $X,Y \in \mathfrak X(M)$ . The Lie derivative of Y with respect to X is equal to the Lie bracket [ X , Y ].

One of the most important applications of the Lie derivative is as an obstruction to invariance under a flow. If $\theta$ is a smooth flow, we say that a vector field Y is invariant under ${\varvec{\theta }}$ if $(\theta _t)_* Y = Y$ wherever the left-hand side is defined.

Proposition A.47.

[ LeeSM , Thm. 9.42] Let M be a smooth manifold and $X\in \mathfrak X(M)$ . A smooth vector field is invariant under the flow of X if and only if its Lie derivative with respect to X is identically zero.

A k -tuple of vector fields $X_1,\dots ,X_k$ is said to commute if $$[X_i,X_j]=0$$ for each i and j .

Proposition A.48

(Canonical Form for Commuting Vector Fields). [ LeeSM , Thm. 9.46] Let M be a smooth n -manifold, and let $(X_1,\dots ,X_k)$ be a linearly independent k -tuple of smooth commuting vector fields on an open subset $W\subseteq M$ . For each $p\in W,$ there exists a smooth coordinate chart $\left( U,\left( x^i\right) \right)$ centered at p such that $X_i = \partial /\partial x^i$ for $i=1,\dots , k$ .

Smooth Covering Maps

A covering map is a surjective continuous map $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ between connected and locally path-connected topological spaces, for which each point of M has connected neighborhood U that is evenly covered , meaning that each connected component of $\pi ^{-1}(U)$ is mapped homeomorphically onto U by $\pi$ . It is called a smooth covering map if ${{\widetilde{M}}}$ and M are smooth manifolds with or without boundary and each component of $\pi ^{-1}(U)$ is mapped diffeomorphically onto U . For every evenly covered open set $U\subseteq M$ , the components of $\pi ^{-1}(U)$ are called the sheets of the covering over ${\varvec{U}}$ .

Here are the main properties of covering maps that we need.

Proposition A.49

(Elementary Properties of Smooth Covering Maps). [ LeeSM , Prop. 4.33]

(a)
Every smooth covering map is a local diffeomorphism, a smooth submersion, an open map, and a quotient map.
(b)
An injective smooth covering map is a diffeomorphism.
(c)
A topological covering map is a smooth covering map if and only if it is a local diffeomorphism.

Proposition A.50

A covering map is a proper map if and only if it is finite-sheeted.

$\blacktriangleright$ Exercise A.51. Prove the preceding proposition.

Example A.52

(A Covering of the ${\varvec{n}}$ -Torus). The n-torus is the manifold $\mathbb T^n=\mathbb {S}^1\times \cdots \times \mathbb {S}^1$ , regarded as the subset of $\mathbb {R}^{2n}$ defined by $(x^1)^2 + (x^2)^2 = \dots = (x^{2n-1})^2 + (x^{2n})^2 = 1$ . The smooth map $X:\mathbb {R}^n\mathrel {\rightarrow }\mathbb T^n$ given by $X(u^1,\dots ,u^n) = (\cos u^1,\sin u^1,\dots ,\cos u^n,\sin u^n)$ is a smooth covering map. //

If $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a covering map and $F:B\mathrel {\rightarrow }M$ is a continuous map from a topological space B into M , then a lift of ${\varvec{F}}$ is a continuous map $\tilde{F}:B\mathrel {\rightarrow }{{\widetilde{M}}}$ such that $\pi \circ \tilde{F} = F$ .

Proposition A.53

(Lifts of Smooth Maps are Smooth). If $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a smooth covering map, B is a smooth manifold with or without boundary, and $F:B\mathrel {\rightarrow }M$ is a smooth map, then every lift of F is smooth.

Proof.

Since $\pi$ is a smooth submersion, every lift $\tilde{F}:B\mathrel {\rightarrow }{{\widetilde{M}}}$ can be written locally as $\tilde{F} = \sigma \circ F$ , where $\sigma$ is a smooth local section of $\pi$ (see Thm. A.17 ). $\square$

Proposition A.54

(Lifting Properties of Covering Maps). Suppose $\pi :{{{\widetilde{M}}}\mathrel {\rightarrow }M}$ is a covering map.

(a)
Unique Lifting Property [ LeeTM , Thm. 11.12] If B is a connected topological space and $F:B\mathrel {\rightarrow }M$ is a continuous map, then any two lifts of F that agree at one point are identical.
(b)
Path Lifting Property [ LeeTM , Cor. 11.14] Suppose $f:[0, 1]\mathrel {\rightarrow }M$ is a continuous path. For every $\tilde{p}\in \pi ^{-1}(f(0))$ , there exists a unique lift $\tilde{f}:[0, 1]\mathrel {\rightarrow }{{\widetilde{M}}}$ of f such that $\tilde{f}(0) = \tilde{p}$ .
(c)
Monodromy Theorem [ LeeTM , Thm. 11.15] Suppose $f,g:[0, 1]\mathrel {\rightarrow }M$ are path-homotopic paths and $\tilde{f},\tilde{g}:[0, 1]\mathrel {\rightarrow }{{\widetilde{M}}}$ are their lifts starting at the same point. Then $\tilde{f}$ and $\tilde{g}$ are path-homotopic and $\tilde{f}(1) = \tilde{g}(1)$ .

Theorem A.55

(Injectivity Theorem). [ LeeTM , Thm. 11.16] If $\pi :{{{\widetilde{M}}}\mathrel {\rightarrow }M}$ is a covering map, then for each point ${\tilde{x}}\in {{\widetilde{M}}}$ , the induced fundamental group homomorphism $\pi _*:\pi _1\big ({{\widetilde{M}}},{\tilde{x}}\big )\mathrel {\rightarrow }\pi _1\big (M,\pi ({\tilde{x}})\big )$ is injective.

Theorem A.56

(Lifting Criterion). [ LeeTM , Thm. 11.18] Suppose $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a covering map, B is a connected and locally path-connected topological space, and $F:B\mathrel {\rightarrow }M$ is a continuous map. Given $b\in B$ and $\tilde{x}\in {{\widetilde{M}}}$ such that $\pi \big (\tilde{x}\big ) = F(b)$ , the map F has a lift to ${{\widetilde{M}}}$ if and only if $F_*\big (\pi _1(B,b)\big )\subseteq \pi _*\big (\pi _1\big ({{\widetilde{M}}},{\tilde{x}}\big )\big )$ .

Corollary A.57

(Lifting Maps from Simply Connected Spaces). [ LeeTM , Cor. 11.19] Suppose $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ and $F:B\mathrel {\rightarrow }M$ satisfy the hypotheses of Theorem A.56 , and in addition B is simply connected. Then every continuous map $F:B\mathrel {\rightarrow }M$ has a lift to ${{\widetilde{M}}}$ . Given any $b\in B$ , the lift can be chosen to take b to any point in the fiber over F ( b ).

Corollary A.58

(Covering Map Homeomorphism Criterion). A covering map $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a homeomorphism if and only if the induced homomorphism $\pi _*:\pi _1\big ({{\widetilde{M}}},\tilde{x}\big )\mathrel {\rightarrow }\pi _1(M,\pi (\tilde{x}))$ is surjective for some (hence every) $\tilde{x}\in {{\widetilde{M}}}$ . A smooth covering map is a diffeomorphism if and only if the induced homomorphism is surjective.

Proof.

By Theorem A.56 , the hypothesis implies that the identity map $\mathrm{Id}:M\mathrel {\rightarrow }M$ has a lift $\widetilde{\mathrm{Id}}:M\mathrel {\rightarrow }{{\widetilde{M}}}$ , which in this case is a continuous inverse for $\pi$ . If $\pi$ is a smooth covering map, then the lift is also smooth. $\square$

Corollary A.59

(Coverings of Simply Connected Spaces). [ LeeTM , Cor. 11.33] If M is a simply connected manifold with or without boundary, then every covering of M is a homeomorphism, and if M is smooth, every smooth covering is a diffeomorphism.

Proposition A.60

(Existence of a Universal Covering Manifold). [ LeeSM , Cor. 4.43] If M is a connected smooth manifold, then there exist a simply connected smooth manifold ${{\widetilde{M}}}$ , called the universal covering manifold of ${\varvec{M}}$ , and a smooth covering map $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ . It is unique in the sense that if ${{\widetilde{M}}}'$ is any other simply connected smooth manifold that admits a smooth covering map $\pi ':{{\widetilde{M}}}'\mathrel {\rightarrow }M$ , then there exists a diffeomorphism $\varPhi :{{\widetilde{M}}}\mathrel {\rightarrow }{{\widetilde{M}}}'$ such that $\pi '\circ \varPhi = \pi$ .

Proposition A.61.

[ LeeTM , Cor. 11.31] With $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ as in the previous proposition, each fiber of $\pi$ has the same cardinality as the fundamental group of M .

$\blacktriangleright$ Exercise A.62. Suppose $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a covering map. Show that ${{\widetilde{M}}}$ is compact if and only if M is compact and $\pi$ is a finite-sheeted covering.

We will revisit smooth covering maps at the end of Appendix C .

Appendix B

Review of Tensors

Abstract Of all the constructions in smooth manifold theory, the ones that play the most fundamental roles in Riemannian geometry are tensors and tensor fields .This appendix offers a brief review of their definitions and properties.

Of all the constructions in smooth manifold theory, the ones that play the most fundamental roles in Riemannian geometry are tensors and tensor fields . Most of the technical machinery of Riemannian geometry is built up using tensors; indeed, Riemannian metrics themselves are tensor fields. This appendix offers a brief review of their definitions and properties. For a more detailed exposition of the material summarized here, see [ LeeSM ].

Tensors on a Vector Space

We begin with tensors on a finite-dimensional vector space. There are many ways of constructing tensors on a vector space, but for simplicity we will stick with the most concrete description, as real-valued multilinear functions. The simplest tensors are just linear functionals, also called covectors .

Covectors

Let V be an n -dimensional vector space (all of our vector spaces are assumed to be real). When we work with bases for V , it is usually important to consider ordered bases , so will assume that each basis comes endowed with a specific ordering. We use parentheses to denote ordered k -tuples, and braces to denote unordered ones, so an ordered basis is designated by either $(b_1,\dots ,b_n)$ or $$(b_i)$$ , and the corresponding unordered basis by $\{b_1,\dots ,b_n\}$ or $\{b_i\}$ .

The dual space of ${\varvec{V}}$ , denoted by $$V^*$$ , is the space of linear maps from V to $\mathbb {R}$ . Elements of the dual space are called covectors or linear functionals on ${\varvec{V}}$ . Under the operations of pointwise addition and multiplication by constants, $$V^*$$ is a vector space.

Suppose $(b_1,\dots ,b_n)$ is an ordered basis for V . For each $i=1,\dots ,n$ , define a covector $\beta ^i\in V^*$ by

$\begin{aligned} \beta ^i(b_j) = \delta ^i_j, \end{aligned}$

where $\delta ^i_j$ is the Kronecker delta symbol , defined by

$\begin{aligned} \delta ^i_{j} = \left\{ \begin{array}{l@{\quad }l} 1 &{}\text { if}~i=j,\\ 0 &{}\text { if}~i\ne j. \end{array}\right. \end{aligned}$

(B.1)

It is a standard exercise to prove that $\big (\beta ^1,\dots ,\beta ^n\big )$ is a basis for $$V^*$$

, called the dual basis to ${\varvec{(b_i)}}$ . Therefore, $$V^*$$

is finite-dimensional, and its dimension is the same as that of V . Every covector $\omega \in V^*$ can thus be written in terms of the dual basis as

$\begin{aligned} \omega = \omega _j \beta ^j, \end{aligned}$

(B.2)

where the components $\omega _j$ are defined by $\omega _j = \omega (b_j)$ . The action of $\omega$ on an arbitrary vector $v = v^i b_i\in V$ is then given by

$\begin{aligned} \omega (v) = \omega _j v^j. \end{aligned}$

(B.3)

(Here and throughout the rest of this appendix we use the Einstein summation convention; see p. xx.)

Every vector $v\in V$ uniquely determines a linear functional on $$V^*$$ , by $\omega \mapsto \omega (v)$ . Because we are assuming V to be finite-dimensional, it is straightforward to show that this correspondence gives a canonical (basis-independent) isomorphism between V and $V^{**}$ (the dual space of $$V^*$$ ). Given $\omega \in V^*$ and $v\in V$ , we can denote the natural action of $\omega$ on v either by the traditional functional notation $\omega (v)$ , or by either of the more symmetric notations $\langle \omega ,v\rangle$ or $\langle v,\omega \rangle$ . The latter notations are meant to emphasize that it does not matter whether we think of the resulting number as the effect of the linear functional $\omega$ acting on the vector v , or as the effect of $v\in V^{**}$ acting on the covector $\omega$ . Note that when applied to a vector and a covector, this pairing makes sense without any choice of an inner product on V .

Higher-Rank Tensors on a Vector Space

Now we generalize from linear functionals to multilinear ones. If $V_1,\dots ,V_k$ and W are vector spaces, a map $F:V_1\times \dots \times V_k\mathrel {\rightarrow }W$ is said to be multilinear if it is linear as a function of each variable separately, when all the others are held fixed:

$\begin{aligned} F(v_1,\dots ,a v_i + a' v_i',\dots ,v_k) = a F(v_1,\dots ,v_i,\dots ,v_k) + a' F(v_1,\dots ,v_i',\dots ,v_k). \end{aligned}$

Given a finite-dimensional vector space V , a covariant k-tensor on ${\varvec{V}}$ is a multilinear map

../images/56724_2_En_BookBackmatter_Equ84_HTML.png

Similarly, a contravariant k-tensor on ${\varvec{V}}$ is a multilinear map

../images/56724_2_En_BookBackmatter_Equ85_HTML.png

We often need to consider tensors of mixed types as well. A mixed tensor of type $\mathbf{( {{\varvec{k, l}}}{} \mathbf ) }$ , also called a k-contravariant, l-covariant tensor , is a multilinear map

../images/56724_2_En_BookBackmatter_Equ86_HTML.png

Actually, in many cases it is necessary to consider real-valued multilinear functions whose arguments consist of k covectors and l vectors, but not necessarily in the order implied by the definition above; such an object is still called a tensor of type ( k , l ). For any given tensor, we will make it clear which arguments are vectors and which are covectors.

The spaces of tensors on V of various types are denoted by

$\begin{aligned} T^k(V^*)&= \{\text {covariant}~k\text {-tensors on}~V\};\\ T^k(V)&= \{\text {contravariant}~k\text {-tensors on}~V\};\\ T^{(k,l)}(V)&= \{\text {mixed}~(k,l)\text {-tensors on}~V\}. \end{aligned}$

The rank of a tensor is the number of arguments (vectors and/or covectors) it takes. By convention, a 0-tensor is just a real number. (You should be aware that the notation conventions for describing the spaces of covariant, contravariant, and mixed tensors are not universally agreed upon. Be sure to look closely at each author’s conventions.)

Tensor Products

There is a natural product, called the tensor product , linking the various tensor spaces over V : if $F\in T^{(k,l)}(V)$ and $G\in T^{(p,q)}(V)$ , the tensor $F\otimes G\in T^{(k+p,l+q)}(V)$ is defined by

$\begin{aligned}&F\otimes G(\omega ^1,\dots ,\omega ^{k+p}, v_1,\dots ,v_{l+q})\\&\qquad \qquad \qquad = F(\omega ^1,\dots ,\omega ^k,v_1,\dots ,v_l)G(\omega ^{k+1},\dots ,\omega ^{k+p},v_{l+1},\dots ,v_{l+q}). \end{aligned}$

The tensor product is associative, so we can unambiguously form tensor products of three or more tensors on V . If $$(b_i)$$

is a basis for V and $\big (\beta ^j\big )$ is the associated dual basis, then a basis for $T^{(k,l)}(V)$ is given by the set of all tensors of the form

$\begin{aligned} b_{i_1}\mathbin {\otimes }\cdots \mathbin {\otimes }b_{i_k} \mathbin {\otimes }\beta ^{j_1}\mathbin {\otimes }\cdots \mathbin {\otimes }\beta ^{j_l} , \end{aligned}$

(B.4)

as the indices $$i_p$$

range from 1 to n . These tensors act on basis elements by

$\begin{aligned} b_{i_1}\mathbin {\otimes }\cdots \mathbin {\otimes }b_{i_k} \mathbin {\otimes }\beta ^{j_1}\mathbin {\otimes }\cdots \mathbin {\otimes }\beta ^{j_l} (\beta ^{s_1},\dots ,\beta ^{s_k},b_{r_1},\dots ,b_{r_l}) = \delta ^{s_1}_{i_1}\cdots \delta ^{s_k}_{i_k} \delta ^{j_1}_{r_1}\cdots \delta ^{j_l}_{r_l} . \end{aligned}$

It follows that $T^{(k,l)}(V)$ has dimension $n^{k+l}$ , where $n=\dim V$ . Every tensor $F\in T^{(k,l)}(V)$ can be written in terms of this basis (using the summation convention) as

$\begin{aligned} F = F_{j_1\dots j_l}^{i_1\dots i_k}\, b_{i_1}\mathbin {\otimes }\cdots \mathbin {\otimes }b_{i_k} \mathbin {\otimes }\beta ^{j_1}\mathbin {\otimes }\cdots \mathbin {\otimes }\beta ^{j_l} , \end{aligned}$

(B.5)

where

$\begin{aligned} F_{j_1\dots j_l}^{i_1\dots i_k} = F\big (\beta ^{i_1},\dots , \beta ^{i_k},b_{j_1},\dots , b_{j_l}\big ). \end{aligned}$

If the arguments of a mixed tensor F occur in a nonstandard order, then the horizontal as well as vertical positions of the indices are significant and reflect which arguments are vectors and which are covectors. For example, if A is a (1, 2)-tensor whose first argument is a vector, second is a covector, and third is a vector, its basis expression would be written

$\begin{aligned} A = A { {A}}_{ {ik}i}{ {A}}^{ {j}j} { {A}}_{ {ik}k} \,\beta ^i\otimes b_j \otimes \beta ^k, \end{aligned}$

where

$\begin{aligned} A { {A}}_{ {ik}i}{ {A}}^{ {j}j} { {A}}_{ {ik}k} = A\big (b_i,\beta ^j,b_k\big ). \end{aligned}$

(B.6)

There are obvious identifications among some of these tensor spaces:

$\begin{aligned} \begin{aligned} T^{(0,0)}(V)&= T^0(V) = T^0(V^*) =\mathbb {R},\\ T^{(1,0)}(V)&= T^1(V) = V,\\ T^{(0,1)}(V)&= T^1(V^*) = V^*,\\ T^{(k,0)}(V)&= T^k(V),\\ T^{(0,k)}(V)&= T^k(V^*). \end{aligned} \end{aligned}$

(B.7)

A less obvious, but extremely important, identification is the following:

$\begin{aligned} T^{(1,1)}(V)\cong \mathrm{End}(V), \end{aligned}$

where $\mathrm{End}(V)$ denotes the space of linear maps from V to itself (also called the endomorphisms of ${\varvec{V}}$ ). This is a special case of the following proposition.

Proposition B.1.

Let V be a finite-dimensional vector space. There is a natural (basis-independent) isomorphism between $T^{(k+1,l)}(V)$ and the space of multilinear maps

../images/56724_2_En_BookBackmatter_Equ87_HTML.png

$\blacktriangleright$ Exercise B.2. Prove Proposition B.1 . [Hint: In the special case $$k=0$$ , $$l=1$$ , consider the map $\varPhi :\mathrm{End}(V)\mathrel {\rightarrow }T^{(1,1)}(V)$ defined by letting $\varPhi A$ be the (1, 1)-tensor defined by $\varPhi A(\omega ,v) = \omega (Av)$ . The general case is similar.]

We can use the result of Proposition B.1 to define a natural operation called trace or contraction , which lowers the rank of a tensor by 2. In one special case, it is easy to describe: the operator ${{\,\mathrm{tr}\,}}:T^{(1,1)}(V)\mathrel {\rightarrow }\mathbb {R}$ is just the trace of F when it is regarded as an endomorphism of V , or in other words the sum of the diagonal entries of any matrix representation of F . Since the trace of a linear endomorphism is basis-independent, this is well defined. More generally, we define ${{\,\mathrm{tr}\,}}:T^{(k+1,l+1)}(V)\mathrel {\rightarrow }T^{(k,l)}(V)$ by letting $({{\,\mathrm{tr}\,}}F)(\omega ^1,\dots ,\omega ^k,v_1,\dots ,v_l)$ be the trace of the (1, 1)-tensor

$\begin{aligned} F\big (\omega ^1,\dots ,\omega ^k,\, \cdot \, ,v_1,\dots ,v_l,\, \cdot \,\big )\in T^{(1,1)}(V). \end{aligned}$

In terms of a basis, the components of ${{\,\mathrm{tr}\,}}F$ are

$\begin{aligned} ({{\,\mathrm{tr}\,}}F)^{i_1\dots i_k}_{j_1\dots j_l} = F^{i_1\dots i_k m}_{j_1\dots j_l m}. \end{aligned}$

In other words, just set the last upper and lower indices equal and sum. Even more generally, we can contract a given tensor on any pair of indices as long as one is contravariant and one is covariant. There is no general notation for this operation, so we just describe it in words each time it arises. For example, we can contract the tensor A with components given by ( B.6 ) on its first and second indices to obtain a covariant 1-tensor B whose components are $B_k= A { {A}}_{ {ik}i}{ {A}}^{ {i}i}{ {A}}_{ {ik}k}$ .

$\blacktriangleright$ Exercise B.3. Show that the trace on any pair of indices (one upper and one lower) is a well-defined linear map from $T^{(k+1,l+1)}(V)$ to $T^{(k,l)}(V)$ .

Symmetric Tensors

There are two classes of tensors that play particularly important roles in differential geometry: the symmetric and alternating tensors. Here we discuss the symmetric ones; we will take up alternating tensors later in this appendix when we discuss differential forms.

If V is a finite-dimensional vector space, a covariant tensor $F\in T^k(V^*)$ is said to be symmetric if its value is unchanged by interchanging any pair of arguments:

$\begin{aligned} F(v_1,\dots ,v_i,\dots ,v_j,\dots ,v_k) = F(v_1,\dots ,v_j,\dots ,v_i,\dots ,v_k) \end{aligned}$

whenever $1\le i<j\le k$ . It follows immediately that the value is unchanged under every rearrangement of the arguments. If we denote the components of F with respect to a basis by $F_{i_1\dots i_k}$ , then F is symmetric if and only if its components are unchanged by every permutation of the indices $i_1,\dots , i_k$ . Every 0-tensor and every 1-tensor are vacuously symmetric. The set of symmetric k -tensors on V is a linear subspace of $$T^k(V^*)$$

, which we denote by $\Sigma ^k(V^*)$ .

A tensor product of symmetric tensors is generally not symmetric. However, there is a natural product of symmetric tensors that does yield symmetric tensors. Given a covariant k -tensor F , the symmetrization of ${\varvec{F}}$ is the k - tensor ${\mathrm{Sym}}~F$ defined by

$\begin{aligned} (\mathrm{Sym}F)(v_1,\dots ,v_k)= \frac{1}{k!}\sum _{\sigma \in S_k} F\left( v_{\sigma (1)},\dots , v_{\sigma (k)}\right) , \end{aligned}$

where

is the group of all permutations of $\{1,\dots ,k\}$ , called the symmetric group on ${\varvec{k}}$ elements . It is easy to check that $\mathrm{Sym}F$ is symmetric, and $\mathrm{Sym}F=F$ if and only if F is symmetric. Then if F and G are symmetric tensors on V , of ranks k and l , respectively, their symmetric product is defined to be the $$(k+l)$$

- tensor FG (denoted by juxtaposition without an explicit product symbol) given by

$\begin{aligned} FG = \mathrm{Sym}(F\otimes G). \end{aligned}$

The most important special case is the symmetric product of two 1-tensors, which can be characterized as follows: if $\omega$ and $\eta$ are covectors, then

$\begin{aligned} \omega \eta = \tfrac{1}{2} (\omega \otimes \eta + \eta \otimes \omega ). \end{aligned}$

(To prove this, just evaluate both sides on an arbitrary pair of vectors ( v , w ), and use the definition of $\omega \eta$ .) If $\omega$ is any 1-tensor, the notation $\omega ^2$ means the symmetric product $\omega \omega$ , which in turn is equal to $\omega \otimes \omega$ .

Tensor Bundles and Tensor Fields

On a smooth manifold M with or without boundary, we can perform the same linear-algebraic constructions on each tangent space $$T_pM$$ that we perform on any vector space, yielding tensors at p . The disjoint union of tensor spaces of a particular type at all points of the manifold yields a vector bundle, called a tensor bundle .

The most fundamental tensor bundle is the cotangent bundle , defined as

$\begin{aligned} T^*M = \coprod _{p\in M} T_p^*M. \end{aligned}$

More generally, the bundle of (k,l)-tensors on M is defined as

$\begin{aligned} T^{(k,l)}TM = \coprod _{p\in M}T^{(k,l)}(T_pM). \end{aligned}$

As special cases, the bundle of covariant k-tensors is denoted by $T^k T^*M = T^{(0,k)}TM$ , and the bundle of contravariant k-tensors is denoted by $T^k TM = T^{(k,0)}TM$ . Similarly, the bundle of symmetric k-tensors is

$\begin{aligned} \Sigma ^k T^*M = \coprod _{p\in M}\Sigma ^k(T^*_pM). \end{aligned}$

There are the usual identifications among these bundles that follow from ( B.7 ): for example, $T^1TM = T^{(1,0)}TM = TM$ and $T^1T^*M = T^{(0,1)}TM = \Sigma ^1T^*M = T^*M$ .

$\blacktriangleright$ Exercise B.4. Show that each tensor bundle is a smooth vector bundle over M , with a local trivialization over every open subset that admits a smooth local frame for TM .

A tensor field on ${\varvec{M}}$ is a section of some tensor bundle over M (see p. xx for the definition of a section of a bundle). A section of $T^1 T^*M = T^{(0,1)}TM$ (a covariant 1-tensor field) is also called a covector field . As we do with vector fields, we write the value of a tensor field F at $p\in M$ as $$F_p$$

. Because covariant tensor fields are the most common and important tensor fields we work with, we use the following shorthand notation for the space of all smooth covariant k -tensor fields:

$\begin{aligned} \mathscr {T}^k(M)&= \varGamma \big ( T^kT^*M\big ). \end{aligned}$

The space of smooth 0-tensor fields is just $C^\infty (M)$ .

Let $(E_i) = (E_1,\dots ,E_n)$ be any smooth local frame for TM over an open subset $U\subseteq M$ . Associated with such a frame is the dual coframe , which we typically denote by $(\varepsilon ^1,\dots ,\varepsilon ^n)$ ; these are smooth covector fields satisfying $\varepsilon ^i(E_j) = \smash {\delta ^i_j}$ . For example, given a coordinate frame $\big (\partial /\partial x^1,\dots ,\partial /\partial x^n\big )$ over some open subset $U\subseteq M$ , the dual coframe is $\big ( dx^1,\dots ,dx^n\big )$ , where $$dx^i$$ is the differential of the coordinate function $$x^i$$ .

In terms of any smooth local frame $$(E_i)$$

and its dual coframe $(\varepsilon ^i)$ , the tensor fields ${E_{i_1}\mathbin {\otimes }\cdots \mathbin {\otimes }E_{i_k} \mathbin {\otimes }\varepsilon ^{j_1}\mathbin {\otimes }\cdots \mathbin {\otimes }\varepsilon ^{j_l}}$ form a smooth local frame for $T^{(k,l)}(T^*M)$ . In particular, in local coordinates $\big (x^i\big )$ , a ( k , l )-tensor field F has a coordinate expression of the form

$\begin{aligned} F = F_{j_1\dots j_l}^{i_1\dots i_k}\, \partial _{i_1}\mathbin {\otimes }\cdots \mathbin {\otimes }\partial _{i_k} \mathbin {\otimes }dx^{j_1}\mathbin {\otimes }\cdots \mathbin {\otimes }dx^{j_l}, \end{aligned}$

(B.8)

where each coefficient $F_{j_1\dots j_l}^{i_1\dots i_k}$ is a smooth real-valued function on U .

$\blacktriangleright$ Exercise B.5. Suppose $F:M\mathrel {\rightarrow }T^{(k,l)} TM$ is a rough ( k , l )-tensor field. Show that F is smooth on an open set $U\subseteq M$ if and only if whenever $\omega ^1,\dots ,\omega ^k$ are smooth covector fields and $X_1,\dots ,X_l$ are smooth vector fields defined on U , the real-valued function ${F\left( \omega ^1,\dots ,\omega ^k,X_1,\dots ,X_l\right) }$ , defined on U by

$\begin{aligned} F\left( \omega ^1,\dots ,\omega ^k,X_1,\dots ,X_l\right) (p) = F_p\left( \omega ^1|_p,\dots ,\omega ^k|_p,X_1|_p,\dots ,X_l|_p\right) , \end{aligned}$

is smooth.

An important property of tensor fields is that they are multilinear over the space of smooth functions. Suppose $F\in \varGamma \big (T^{(k,l)}TM\big )$ is a smooth tensor field. Given smooth covector fields $\omega ^1,\dots ,\omega ^k\in \mathscr {T}^1(M)$ and smooth vector fields $X_1,\dots ,X_l\in \mathfrak X(M)$ , Exercise B.5 shows that the function $F\left( \omega ^1,\dots ,\omega ^k,X_1,\dots ,X_l\right)$ is smooth, and thus F induces a map

../images/56724_2_En_BookBackmatter_Equ88_HTML.png

It is easy to check that this map is multilinear over ${\varvec{C^\infty (M)}}$ , that is, for all functions $u,v\in C^\infty (M)$ and smooth vector or covector fields $\alpha$ , $\beta$ ,

$\begin{aligned} \tilde{F}(\dots ,u\alpha +v\beta ,\dots ) = u\tilde{F}(\dots ,\alpha ,\dots ) + v\tilde{F} (\dots ,\beta ,\dots ). \end{aligned}$

Even more important is the converse: as the next lemma shows, every such map that is multilinear over $C^\infty (M)$ defines a tensor field. (This lemma is stated and proved in [ LeeSM ] for covariant tensor fields, but the same argument works in the case of mixed tensors.)

Lemma B.6

(Tensor Characterization Lemma). [ LeeSM , Lemma 12.24] A map

../images/56724_2_En_BookBackmatter_Equ89_HTML.png

is induced by a smooth ( k , l )-tensor field as above if and only if it is multilinear over $C^\infty (M)$ . Similarly, a map

../images/56724_2_En_BookBackmatter_Equ90_HTML.png

is induced by a smooth $$(k+1,l)$$

-tensor field as in Proposition B.1 if and only if it is multilinear over $C^\infty (M)$ .

Because of this result, it is common to use the same symbol for both a tensor field and the multilinear map on sections that it defines, and to refer to either of these objects as a tensor field.

Pullbacks of Tensor Fields

Suppose $F:M\mathrel {\rightarrow }N$ is a smooth map and A is a covariant k -tensor field on N . For every $p\in M$ , we define a tensor $dF_p^*(A)\in T^k\bigl (T_p^*M\bigr ),$ called the pointwise pullback of ${\varvec{A}}$ by ${\varvec{F}}$ at ${\varvec{p}}$ , by

$\begin{aligned} dF_p^*(A)(v_1,\dots ,v_k) = A\big (dF_p(v_1),\dots ,dF_p(v_k)\big ) \end{aligned}$

for $v_1,\dots ,v_k\in T_pM$ ; and we define the pullback of ${\varvec{A}}$ by ${\varvec{F}}$ to be the tensor field $$F^*A$$

on M defined by

$\begin{aligned} (F^* A)_p = dF_p^*\big ( A_{F(p)}\big ). \end{aligned}$

Proposition B.7

(Properties of Tensor Pullbacks). [ LeeSM , Prop. 12.25] Suppose $F:M\mathrel {\rightarrow }N$ and $G:P\mathrel {\rightarrow }M$ are smooth maps, A and B are covariant tensor fields on N , and f is a real-valued function on N .

(a)
$F^*(f B) = (f\circ F)F^* B$ .
(b)
$F^*( A\otimes B) = F^* A\otimes F^* B$ .
(c)
.
(d)
is a (continuous) tensor field, and it is smooth if B is smooth.
(e)
$(F\circ G)^* B = G^*( F^* B )$ .
(f)
$(\mathrm{Id}_N)^* B= B$ .

If f is a continuous real-valued function (i.e., a 0-tensor field), the pullback $$F^*f$$ is just the composition $f\circ F$ .

The following proposition shows how pullbacks are computed in local coordinates.

Proposition B.8.

[ LeeSM , Cor. 12.28] Let $F:M\mathrel {\rightarrow }N$ be smooth, and let B be a covariant k -tensor field on N . If $p\in M$ and $\bigl (y^i\bigr )$ are smooth coordinates for N on a neighborhood of F ( p ), then $$F^* B$$

has the following expression in a neighborhood of p :

$\begin{aligned} F^*\left( B_{i_1\dots i_k}dy^{i_1}\otimes \cdots \otimes dy^{i_k}\right) = \left( B_{i_1\dots i_k}\circ F\right) d\!\left( y^{i_1}\circ F\right) \otimes \cdots \otimes d\!\left( y^{i_k}\circ F\right) . \end{aligned}$

Lie Derivatives of Tensor Fields

We can extend the notion of Lie derivatives to tensor fields. This can be done for mixed tensor fields of any rank, but for simplicity we restrict attention to covariant tensor fields. Suppose X is a smooth vector field on M and $\theta$ is its flow. If A is a smooth covariant tensor field on M , the Lie derivative of ${\varvec{A}}$ with respect to ${\varvec{X}}$ is the smooth covariant tensor field $\mathscr {L}_X A$ defined by

$\begin{aligned} \left( \mathscr {L}_X A\right) _p = \left. \frac{d}{dt}\right| _{t=0} (\theta _t^*A)_p = \lim _{t\mathrel {\rightarrow }0} \frac{d(\theta _t)_{p}^*\bigl (A_{\theta _t(p)}\bigr ) - A_p}{t}. \end{aligned}$

Proposition B.9.

[ LeeSM , Thm. 12.32] Suppose M is a smooth manifold, $X \in \mathfrak X(M)$ , and A is a smooth covariant k -tensor field on M . For all smooth vector fields $Z_1,\dots ,Z_k$ ,

$\begin{aligned}&(\mathscr {L}_X A)(Z_1,\dots ,Z_k) = X\big ( A(Z_1,\dots ,Z_k) \big ) \\&\quad - A(\mathscr {L}_X Z_1,Z_2,\dots ,Z_k) - A(Z_1,\mathscr {L}_X Z_2,\dots ,Z_k) - \dots - A(Z_1,\dots ,\mathscr {L}_X Z_k). \end{aligned}$

As we did for vector fields, we say that a covariant tensor field A is invariant under a flow ${\varvec{\theta }}$ if $(\theta _t)^* A = A$ wherever it is defined. The next proposition is a tensor analogue of Proposition A.47 .

Proposition B.10.

[ LeeSM , Thm. 12.37] Let M be a smooth manifold and $X\in \mathfrak X(M)$ . A smooth covariant tensor field is invariant under the flow of X if and only if its Lie derivative with respect to X is identically zero.

Differential Forms and Integration

In addition to symmetric tensors, the other class of tensors that play a special role in differential geometry is that of alternating tensors , which we now define. Let V be a finite-dimensional vector space. If F is a covariant k -tensor on V , we say that F is alternating if its value changes sign whenever two different arguments are interchanged:

$\begin{aligned} F(v_1,\dots ,v_i,\dots ,v_j,\dots ,v_k) = - F(v_1,\dots ,v_j,\dots ,v_i,\dots ,v_k). \end{aligned}$

The set of alternating covariant k -tensors on V is a linear subspace of $$T^k(V^*)$$

, denoted by $\Lambda ^k(V^*)$ . If $F\in \Lambda ^k(V^*)$ , then the effect of an arbitrary permutation of its arguments is given by

$\begin{aligned} F(v_{\sigma (1)},\dots ,v_{\sigma (k)}) = ({\mathrm{sgn}}~\sigma ) F(v_{1},\dots ,v_{k}), \end{aligned}$

where ${\mathrm{sgn}}~\sigma$ represents the sign of the permutation $\sigma \in S_k$ , which is $$+1$$

if $\sigma$ is even (i.e., can be written as a composition of an even number of transpositions), and $$-1$$

if $\sigma$ is odd. The components of F with respect to any basis change sign similarly under a permutation of the indices. All 0-tensors and 1-tensors are alternating.

Analogously to the symmetrization operator defined above, if F is any covariant k -tensor on V , we define the alternation of ${\varvec{F}}$ by

$\begin{aligned} ({{\,\mathrm{Alt}\,}}F)(v_1,\dots ,v_k) = \frac{1}{k!}\sum _{\sigma \in S_k} ({\mathrm{sgn}}~\sigma ) F\left( v_{\sigma (1)},\dots ,v_{\sigma (k)}\right) . \end{aligned}$

(B.9)

Again, it is easy to check that ${{\,\mathrm{Alt}\,}}F$ is alternating, and ${{\,\mathrm{Alt}\,}}F=F$ if and only if F is alternating. Given $\omega \in \Lambda ^k(V^*)$ and $\eta \in \Lambda ^l(V^*)$ , we define their wedge product by

$\begin{aligned} \omega \wedge \eta =\frac{(k+l)!}{k!l!}{{\,\mathrm{Alt}\,}}(\omega \otimes \eta ). \end{aligned}$

It is immediate that $\omega \wedge \eta$ is an alternating $$(k+l)$$

-tensor. The wedge product is easily seen to be bilinear and anticommutative , which means that

$\begin{aligned} \omega \wedge \eta = (-1)^{kl}\eta \wedge \omega , \qquad \text {for }\omega \in \Lambda ^k(V^*)\text { and }\eta \in \Lambda ^l(V^*). \end{aligned}$

It is also the case, although not so easy to see, that it is associative; see [ LeeSM , Prop. 14.11] for a proof. If $$(b_i)$$

is a basis for V and $(\beta ^i)$ is the dual basis, the wedge products $\beta ^{i_1}\wedge \dots \wedge \beta ^{i_k}$ , as $(i_1,\dots ,i_k)$ range over strictly increasing multi-indices, form a basis for $\Lambda ^k(V^*)$ , which therefore has dimension $\left( {\begin{array}{c}n\\ k\end{array}}\right) = n!/\big (k!(n-k)!\big )$ , where $n=\dim V$ . In terms of these basis elements, the wedge product satisfies

$\begin{aligned} \beta ^{i_1}\wedge \dots \wedge \beta ^{i_k} ( b_{j_1},\dots ,b_{j_k}) = \det \big (\delta ^{i_p}_{j_q}\big ). \end{aligned}$

More generally, if $\omega ^1,\dots ,\omega ^k$ are any covectors and $v_1,\dots ,v_k$ are any vectors, then

$\begin{aligned} \omega ^{1} \wedge \dots \wedge \omega ^k ( v_1,\dots ,v_k) = \det \big (\omega ^i(v_j)\big ). \end{aligned}$

(B.10)

The wedge product can be defined analogously for contravariant alternating tensors $\alpha \in \Lambda ^p(V)$ , $\beta \in \Lambda ^q(V)$ , simply by regarding $\alpha$ , $\beta$ , and $\alpha \wedge \beta$ as alternating multilinear functionals on $$V^*$$

The convention we use for the wedge product is referred to in [ LeeSM ] as the determinant convention . There is another convention that is also in common use, the Alt convention , which amounts to multiplying the right-hand side of ( B.10 ) by a factor of 1 / k !. The choice of which definition to use is a matter of taste, though there are various reasons to justify each choice depending on the context. The determinant convention is most common in introductory differential geometry texts, and is used, for example, in [ Boo86 , Cha06 , dC92 , LeeJeff09 , LeeSM , Pet16 , Spi79 , Tu11 ]. The Alt convention is used in [ KN96 ] and is more common in complex differential geometry.

Given an alternating k -tensor $\omega \in \Lambda ^k(V^*)$ and a vector $v\in V$ , we define an alternating $$(k-1)$$

- tensor ../images/56724_2_En_BookBackmatter_IEq1031_HTML.gif

../images/56724_2_En_BookBackmatter_Equ91_HTML.png

The operation ../images/56724_2_En_BookBackmatter_IEq1032_HTML.gif

is known as interior multiplication by ${\varvec{v}}$ , and is also denoted by $\omega \mapsto i_v\omega$ . By convention, ../images/56724_2_En_BookBackmatter_IEq1036_HTML.gif

when $\omega$ is a 0-tensor. Direct computation shows that $i_v\circ i_v = 0$ , and $$i_v$$

satisfies the following product rule for an alternating k -tensor $\omega$ and an alternating l -tensor $\eta$ :

$\begin{aligned} i_v(\omega \wedge \eta ) = (i_v\omega )\wedge \eta + (-1)^k\omega \wedge (i_v\eta ). \end{aligned}$

(B.11)

If M is a smooth manifold with or without boundary, the subbundle of $$T^kT^*M$$

consisting of alternating tensors is denoted by $\Lambda ^kT^*M$ , and an alternating tensor field on M is called a differential k-form , or just a k-form . The space of all smooth k -forms is denoted by $\Omega ^k(M) = \varGamma \big ( \Lambda ^kT^*M\big )$ . Because every 1-tensor field is alternating, $\Omega ^1(M)$ is the same as the space $\mathscr {T}^1(M)$ of smooth covector fields.

Exterior Derivatives

The most important operation on differential forms is the exterior derivative , defined as follows. Suppose M is a smooth n -manifold with or without boundary, and $\big (x^i\big )$ are any smooth local coordinates on M . A smooth k -form $\omega$ can be expressed in these coordinates as

$\begin{aligned} \omega = \sum _{j_1<\dots <j_k} \omega _{j_1\dots j_k} \,dx^{j_1}\wedge \dots \wedge dx^{j_k}, \end{aligned}$

and then we define the exterior derivative of ${\varvec{\omega }}$ , denoted by $d\omega$ , to be the $$(k+1)$$

-form defined in coordinates by

$\begin{aligned} d\omega = \sum _{j_1<\dots <j_k}\ \sum _{i=1}^n\frac{\partial \omega _{j_1\dots j_k}}{\partial x^i}\, dx^i\wedge dx^{j_1}\wedge \dots \wedge dx^{j_k}. \end{aligned}$

For a 0-form f (a smooth real-valued function), this reduces to

$\begin{aligned} df = \sum _{i=1}^n \frac{\partial f}{\partial x^i} dx^i, \end{aligned}$

which is exactly the differential of f defined by ( A.4 ).

Proposition B.11

(Properties of Exterior Differentiation). [ LeeSM , Thm. 14.24] Suppose M is a smooth n -manifold with or without boundary.

(a)
For each $k=0,\dots , n$ , the operator $d:\Omega ^k(M)\mathrel {\rightarrow }\Omega ^{k+1}(M)$ is well defined, independently of coordinates.
(b)
d is linear over $\mathbb {R}$ .
(c)
$d\circ d\equiv 0$ .
(d)
If $\omega \in \Omega ^k(M)$ and $\eta \in \Omega ^l(M)$ , then
$\begin{aligned} d(\omega \wedge \eta ) = d\omega \wedge \eta + (-1)^k\omega \wedge d\eta . \end{aligned}$

Proposition B.12

(Exterior Derivative of a ${\varvec{1}}$ -form). [ LeeSM , Prop. 14.29] If $\omega$ is a smooth 1-form and X , Y are vector fields, then

$\begin{aligned} d\omega (X,Y) = X (\omega (Y)) - Y (\omega (X)) - \omega ([X,Y]). \end{aligned}$

Proposition B.13

(Naturality of the Exterior Derivative). [ LeeSM , Prop. 14.26] If $F:M\mathrel {\rightarrow }N$ is a smooth map, then for each k , the pullback map $F^*:\Omega ^k(N)\mathrel {\rightarrow }\Omega ^k(M)$ commutes with d : for all $\omega \in \Omega ^k(N)$ ,

$\begin{aligned} F^*(d\omega ) = d(F^*\omega ). \end{aligned}$

(B.12)

A smooth differential form $\omega \in \Omega ^k(M)$ is closed if $d\omega =0$ , and exact if there exists a smooth $$(k-1)$$ -form $\eta$ on M such that $\omega = d\eta$ . The fact that $d\circ d=0$ implies that every exact form is closed, but the converse is not true in general. However, the next lemma gives an important special case in which it is true. If V is a vector space, a subset $S\subseteq V$ is said to be star-shaped with respect to a point $x\in S$ if for every $y\in S$ , the line segment from x to y is contained in S .

Lemma B.14

(The Poincaré Lemma). [ LeeSM , Thm. 17.14] If U is a star-shaped open subset of $\mathbb {R}^n$ , then for $k\ge 1$ , every closed k -form on U is exact.

Orientations

If V is a finite-dimensional vector space, an orientation of ${\varvec{V}}$ is an equivalence class of ordered bases for V , where two ordered bases are considered equivalent if the transition matrix that expresses one basis in terms of the other has positive determinant. Every vector space has exactly two orientations. Once an orientation is chosen, a basis is said to be positively oriented if it belongs to the chosen orientation, and negatively oriented if not. The standard orientation of ${\varvec{\mathbb {R}^n}}$ is the one determined by the standard basis $(e_1,\dots ,e_n)$ , where $$e_i$$ is the vector $(0,\dots ,1,\dots ,0)$ with 1 in the i th place and zeros elsewhere.

If M is a smooth manifold, an orientation for ${\varvec{M}}$ is a choice of orientation for each tangent space that is continuous in the sense that in a neighborhood of every point there is a (continuous) local frame that determines the given orientation at each point of the neighborhood. If there exists an orientation for M , we say that M is orientable . An oriented manifold is a smooth orientable manifold together with a choice of orientation. If M is an oriented n -manifold, then a smooth coordinate chart $\big (U,\big (x^i\big )\big )$ is said to be an oriented chart if the coordinate frame $\big (\partial /\partial x^1,\dots ,\partial /\partial x^n\big )$ is positively oriented at each point. Exactly the same definitions apply to manifolds with boundary.

Proposition B.15

(Orientation Determined by an ${\varvec{n}}$ -Form). [ LeeSM , Prop. 15.5] Let M be an n -manifold with or without boundary. Every nonvanishing n -form $\mu \in \Omega ^n(M)$ determines a unique orientation of M by declaring an ordered basis $(b_1,\dots ,b_n)$ for $$T_pM$$ to be positively oriented if and only if $\mu _p(b_1,\dots ,b_n)>0$ . Conversely, if M is oriented, there is a smooth nonvanishing n -form that determines the orientation in this way.

Because of this proposition, a nonvanishing n -form on a smooth n -manifold is called an orientation form . If in addition M is oriented and $\mu$ determines the given orientation, we say that $\mu$ is positively oriented .

Suppose M and N are both smooth n -manifolds with or without boundary and $F:M\mathrel {\rightarrow }N$ is a local diffeomorphism. If N is oriented, we define the pullback orientation on ${\varvec{M}}$ induced by ${\varvec{F}}$ to be the orientation determined by $F^*\mu$ , where $\mu$ is any positively oriented orientation form for N . If both M and N are oriented, we say that F is orientation-preserving if the pullback orientation is equal to the given orientation on M , and orientation-reversing if the pullback orientation is the opposite orientation.

Proposition B.16

(Orientation of a Hypersurface). [ LeeSM , Prop. 15.21] Suppose M is an oriented smooth n -manifold with or without boundary, $S\subseteq M$ is a smooth immersed hypersurface, and N is a continuous vector field along S (i.e., a continuous map $N:S\mathrel {\rightarrow }TM$ such that $N_p\in T_pM$ for each $p\in S$ ). If N is nowhere tangent to S , then S has a unique orientation determined by the $$(n-1)$$ -form ../images/56724_2_En_BookBackmatter_IEq1104_HTML.gif , where $\iota :S\mathrel {\hookrightarrow }M$ is inclusion and $\mu$ is any positively oriented orientation form for M .

A special case of the preceding proposition occurs when M is a manifold with boundary and the hypersurface in question is the boundary of M . In that case we declare a vector field N along $\partial M$ to be outward-pointing if for each $p\in \partial M$ , there is a smooth curve $\gamma :(-\varepsilon ,0]\mathrel {\rightarrow }M$ such that $\gamma (0)=p$ and $\gamma '(0)=N_p$ , and $N_p\notin T_p(\partial M)$ . We can always construct a global smooth outward-pointing vector field by taking $-\partial /\partial x^n$ in boundary coordinates in a neighborhood of each boundary point, and gluing together with a partition of unity. Since an outward-pointing vector field is nowhere tangent to $\partial M$ , we have the following proposition.

Proposition B.17

(Induced Orientation on a Boundary). [ LeeSM , Prop. 15.24] If M is an oriented smooth manifold with boundary, then $\partial M$ is orientable, and it has a canonical orientation (called the induced orientation or Stokes orientation ) determined by ../images/56724_2_En_BookBackmatter_IEq1116_HTML.gif , where $\iota :\partial M\mathrel {\hookrightarrow }M$ is inclusion, N is any outward-pointing vector field along $\partial M$ , and $\mu$ is any positively oriented orientation form for M .

Proposition B.18

(Orientation Covering Theorem). [ LeeSM , Thm. 15.41] If M is a connected, nonorientable smooth manifold, then there exist an oriented smooth manifold ${\smash [t]{\widehat{M}}}$ and a two-sheeted smooth covering map $\hat{\pi }:{\smash [t]{\widehat{M}}}\mathrel {\rightarrow }M$ , called the orientation covering of ${\varvec{M}}$ .

Corollary B.19.

Every simply connected smooth manifold is orientable.

Proof.

Corollary A.59 shows that a simply connected manifold does not admit two-sheeted covering maps. $\square$

Integration of Differential Forms

Suppose first that $\omega$ is a (continuous) n -form on an open subset $U\subseteq \mathbb {R}^n$ or $\mathbb {R}^n_+$ . It can be written $\omega = f\, dx^1\wedge \dots \wedge dx^n$ for some continuous real-valued function f , and we define the integral of ${\varvec{\omega }}$ over ${\varvec{U}}$ to be the ordinary multiple integral

$\begin{aligned} \int _{U} \omega = \int _{U} f\big (x^1,\dots ,x^n\big )\, dx^1 \cdots dx^n, \end{aligned}$

provided the integral is well defined. This will always be the case, for example, if f is continuous and has compact support in U .

In general, if $\omega$ is a compactly supported n -form on a smooth n -manifold M with or without boundary, we define the integral of ${\varvec{\omega }}$ over ${\varvec{M}}$ by choosing finitely many oriented smooth coordinate charts $\{U_i\}_{i=1}^k$ whose domains cover the support of $\omega$ , together with a smooth partition of unity $\{\psi _1\}_{i=1}^k$ subordinate to this cover, and defining

$\begin{aligned} \int _M \omega = \sum _{i=1}^k \int _{\hat{U}_i} \big (\varphi _i^{-1}\big )^* (\psi _i\omega ), \end{aligned}$

where $\hat{U}_i = \varphi _i(U_i)$ , and the integrals on the right-hand side are defined as above. Proposition 16.5 in [ LeeSM ] shows that this definition does not depend on the choice of oriented charts or partition of unity.

Proposition B.20

(Properties of Integrals of Forms). [ LeeSM , Prop. 16.6] Suppose M and N are nonempty oriented smooth n -manifolds with or without boundary, and $\omega ,\eta$ are compactly supported n -forms on M .

(a)
Linearity: If $a,b\in \mathbb {R}$ , then
$\begin{aligned} \int _M a\omega + b\eta = a\int _M\omega + b\int _M\eta . \end{aligned}$
(b)
Orientation Reversal: If denotes M with the opposite orientation, then
$\begin{aligned} \int _{-M}\omega = - \int _M\omega . \end{aligned}$
(c)
Positivity: If $\omega$ is a positively oriented orientation form, then $\int _M\omega >0$ .
(d)
Diffeomorphism Invariance: If $F:N\mathrel {\rightarrow }M$ is an orientation-preserving or orientation-reversing diffeomorphism, then

Theorem B.21

(Stokes’s Theorem). [ LeeSM , Thm. 16.11] If M is an oriented smooth n -manifold with boundary and $\omega$ is a compactly supported smooth $$(n-1)$$

-form on M , then

$\begin{aligned} \int _M d\omega =\int _{\partial M}\omega . \end{aligned}$

(B.13)

In the statement of this theorem, $\partial M$ is understood to have the Stokes orientation, and the right-hand side is interpreted as the integral of the pullback of $\omega$ to $\partial M$ .

The following special case is frequently useful.

Corollary B.22.

[ LeeSM , Cor. 16.13] Suppose M is a compact oriented smooth n -manifold (without boundary). Then the integral of every exact n -form on M is zero.

Densities

On an oriented n -manifold with or without boundary, n -forms are the natural objects to integrate. But in order to integrate on a nonorientable manifold, we need closely related objects called densities .

If V is an n -dimensional real vector space, a density on ${\varvec{V}}$ is a function

../images/56724_2_En_BookBackmatter_Equ93_HTML.png

satisfying the following formula for every linear map $T:V\mathrel {\rightarrow }V$ :

$\begin{aligned} \mu (Tv_1,\dots ,Tv_n) = \left| \det T\right| \mu (v_1,\dots ,v_n). \end{aligned}$

(B.14)

A density $\mu$ is said to be positive if $\mu (v_1,\dots ,v_n)>0$ whenever $(v_1,\dots ,v_n)$ is a basis of V ; it is clear from ( B.14 ) that if this is true for some basis, then it is true for every one. Every nonzero alternating n -tensor $\mu$ determines a positive density $|\mu |$ by the formula

$\begin{aligned} \left| \mu \right| (v_1,\dots ,v_n) = \left| \mu (v_1,\dots ,v_n)\right| . \end{aligned}$

The set $\mathscr {D}(V)$ of all densities on V is a 1-dimensional vector space, spanned by $|\mu |$ for any nonzero alternating n -tensor $\mu$ .

When M is a smooth manifold with or without boundary, the set

$\begin{aligned} \mathscr {D} M = \coprod _{p\in M} \mathscr {D}(T_pM) \end{aligned}$

is called the density bundle of ${\varvec{M}}$ . It is a smooth rank-1 vector bundle, with $\left| dx^1\wedge \dots \wedge dx^n\right|$ as a smooth local frame over any smooth coordinate chart. A density on ${\varvec{M}}$ is a (smooth) section $\mu$ of $\mathscr {D} M$ ; in any local coordinates, it can be written

$\begin{aligned} \mu = u\, \left| dx^{1}\wedge \cdots \wedge dx^{n}\right| \end{aligned}$

for some locally defined smooth function u .

Under smooth maps, densities pull back in the same way as differential forms: Suppose $F:M\mathrel {\rightarrow }N$ is a smooth map between n -manifolds with or without boundary, and $\mu$ is a density on N . The pullback of ${\varvec{\mu }}$ is the density $F^*\mu$ on M defined by

$\begin{aligned} (F^*\mu )_p(v_1,\dots ,v_n) = \mu _{F(p)}(dF_p(v_1),\dots ,dF_p(v_n)). \end{aligned}$

In coordinates, this satisfies

$\begin{aligned} F^*\left( u \,\left| dy^{1}\wedge \cdots \wedge dy^{n}\right| \right) =(u\circ F)\left| \det DF\right| \,\left| dx^1\wedge \cdots \wedge dx^n\right| , \end{aligned}$

(B.15)

where DF represents the matrix of partial derivatives of F in these coordinates.

Because the pullback formula ( B.15 ) is exactly analogous to the change of variables formula for multiple integrals, we can define the integral of a compactly supported density on a smooth manifold with or without boundary in exactly the same way as the integral of an n -form is defined on an oriented manifold, except that now there is no need to have an orientation. Thus if $\mu = u \,|dx^1\cdots dx^n|$ is a smooth density that is compactly supported in an open set $V\subseteq \mathbb {R}^n$ , we define the integral of $\mu$ by

$\begin{aligned} \int _V \mu = \int _{V} u \,dx^1\cdots dx^n. \end{aligned}$

It follows from ( B.15 ) that the value of this integral is diffeomorphism-invariant, so we can define the integral of a compactly supported density on a smooth manifold with or without boundary by breaking it up with a partition of unity and integrating each term in coordinates as above.

Definitions and Properties

A Lie group is a smooth manifold G that is also a group in the algebraic sense, with the property that the multiplication map $m:G\times G\mathrel {\rightarrow }G$ given by $m(\varphi _1,\varphi _2)=\varphi _1\varphi _2$ and the inversion map $i:G\mathrel {\rightarrow }G$ given by $i(\varphi ) = \varphi ^{-1}$ are smooth. For generic Lie groups, we usually denote the identity element by e , unless there is a more specific common notation for a particular group.

Given a Lie group G , each $\varphi \in G$ defines a map $L_\varphi :G\mathrel {\rightarrow }G$ , called left translation , by $L_\varphi (\varphi ')= \varphi \varphi '$ . It is a diffeomorphism with inverse $L_{\varphi ^{-1}}$ . Similarly, right translation $R_\varphi :G\mathrel {\rightarrow }G$ is the diffeomorphism $R_\varphi (\varphi ') = \varphi '\varphi$ , with inverse $R_{\varphi ^{-1}}$ .

A subgroup $H\subseteq G$ that is endowed with a topology and smooth structure making it into a Lie group and an immersed or embedded submanifold of G is called a Lie subgroup of ${\varvec{G}}$ .

If G and H are Lie groups, a group homomorphism $F:G\mathrel {\rightarrow }H$ that is also a smooth map is called a Lie group homomorphism . It is a Lie group isomorphism if it has an inverse that is also a Lie group homomorphism. A Lie group isomorphism from G to itself is called an automorphism of ${\varvec{G}}$ .

Proposition C.1.

[ LeeSM , Thms. 7.5 & 21.27] Suppose $F:G\mathrel {\rightarrow }H$ is a Lie group homomorphism.

(a)
F has constant rank.
(b)
The kernel of F is an embedded Lie subgroup of G .
(c)
The image of F is an immersed Lie subgroup of H .

Example C.2

(Lie Groups).

(a)
Every countable group with the discrete topology is a zero-dimensional Lie group, called a discrete Lie group .
(b)
$\mathbb {R}^n$ and $\mathbb C^n$ are Lie groups under addition.
(c)
The sets $\mathbb {R}^*$ , $\mathbb {R}^+$ , and $\mathbb C^*$ of nonzero real numbers, positive real numbers, and nonzero complex numbers, respectively, are Lie groups under multiplication.
(d)
The unit circle $\mathbb {S}^1\subseteq \mathbb C$ is a 1-dimensional Lie group under complex multiplication, called the circle group .
(e)
Given Lie groups $G_1,\dots ,G_k,$ their direct product group is the Lie group whose underlying manifold is ${G_1\times \cdots \times G_k}$ , and whose multiplication is given by $(g_1,\dots ,g_k)\left( g_1',\dots ,g_k'\right) = \left( g_1g_1',\dots ,g_kg_k'\right)$ . For example, the n-torus is the n -fold product group $\mathbb T^n = \mathbb {S}^1\times \dots \times \mathbb {S}^1$ . The 2-torus is often simply called the torus .
(f)
The general linear groups ${{\,\mathrm{GL}\,}}(n,\mathbb {R})$ and ${{\,\mathrm{GL}\,}}(n,\mathbb C)$ , consisting of all invertible $n\times n$ real or complex matrices, respectively, are Lie groups under matrix multiplication. The identity element in both cases is the $n\times n$ identity matrix, denoted by . More generally, if V is an n -dimensional real or complex vector space, the group ${{\,\mathrm{GL}\,}}(V)$ of invertible linear maps from V to itself is a Lie group isomorphic to ${{\,\mathrm{GL}\,}}(n,\mathbb {R})$ or ${{\,\mathrm{GL}\,}}(n,\mathbb C)$ .
(g)
The real and complex special linear groups are the subgroups $\mathrm{SL}(n,\mathbb {R})\subseteq {{\,\mathrm{GL}\,}}(n,\mathbb {R})$ and $\mathrm{SL}(n,\mathbb C)\subseteq {{\,\mathrm{GL}\,}}(n,\mathbb C)$ consisting of matrices of determinant 1.
(h)
The orthogonal group ${{\,\mathrm{O}\,}}(n)\subseteq {{\,\mathrm{GL}\,}}(n,\mathbb {R})$ is the subgroup consisting of orthogonal matrices, those that satisfy , where is the transpose of A . The special orthogonal group is the subgroup ${{\,\mathrm{SO}\,}}(n) = {{\,\mathrm{O}\,}}(n)\cap \mathrm{SL}(n,\mathbb {R})$ .
(i)
Similarly, the unitary group $\mathrm{U}(n)\subseteq {{\,\mathrm{GL}\,}}(n,\mathbb C)$ is the subgroup consisting of complex unitary matrices, those that satisfy , where $A^*= \bar{A}{\;}^T$ is the conjugate transpose of A , called its adjoint . The special unitary group is $\mathrm{SU}(n) = \mathrm{U}(n)\cap \mathrm{SL}(n,\mathbb C)$ .//

If G is a Lie group and V is a finite-dimensional vector space, a Lie group homomorphism $\rho :G \mathrel {\rightarrow }{{\,\mathrm{GL}\,}}(V)$ is called a (finite-dimensional) representation of ${\varvec{G}}$ . It is said to be faithful if it is injective.

The Lie Algebra of a Lie Group

Suppose G is a Lie group. A vector field X on G is said to be left-invariant if it is invariant under all left translations, meaning that $$(L_g)_*X=X$$ for every $g\in G$ . Similarly, X is right-invariant if it is invariant under all right translations.

We denote the set of all smooth left-invariant vector fields on G by $\mathrm{Lie}(G)$ . It is a vector subspace of $\mathfrak X(G)$ that is closed under Lie brackets (see [ LeeSM , Prop. 8.33]). Thus it is an example of a Lie algebra : a vector space $\mathfrak g$ endowed with a bilinear operation $[ \cdot , \cdot ]:\mathfrak g\times \mathfrak g\mathrel {\rightarrow }\mathfrak g$ , called the bracket , that is antisymmetric (meaning $$[X,Y] = -[Y,X]$$

) and satisfies the Jacobi identity :

$\begin{aligned}{}[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0. \end{aligned}$

With the Lie bracket structure that it inherits from $\mathfrak X(G)$ , the Lie algebra $\mathrm{Lie}(G)$ is called the Lie algebra of ${\varvec{G}}$ .

Proposition C.3.

[ LeeSM , Thm. 8.37] Let G be a Lie group with identity e . The evaluation map $X\mapsto X_e$ is a vector space isomorphism from $\mathrm{Lie}(G)$ to $$T_eG$$ , so $\mathrm{Lie}(G)$ has the same dimension as G itself.

If $\mathfrak g$ is a Lie algebra, a Lie subalgebra of $\mathfrak g$ is a vector subspace $\mathfrak h\subseteq \mathfrak g$ that is closed under Lie brackets, and is thus a Lie algebra with the restriction of the same bracket operation. Given Lie algebras $\mathfrak g_1$ and $\mathfrak g_2$ , a Lie algebra homomorphism is a linear map $F:\mathfrak g_1\mathrel {\rightarrow }\mathfrak g_2$ that preserves brackets: $$F([X,Y]) = [F(X),F(Y)]$$ . A Lie algebra isomorphism is an invertible Lie algebra homomorphism, and a Lie algebra automorphism is a Lie algebra isomorphism from a Lie algebra to itself.

Example C.4

(Lie Algebras).

(a)
If M is a smooth positive-dimensional manifold, the space $\mathfrak X(M)$ of all smooth vector fields on M is an infinite-dimensional Lie algebra under the Lie bracket.
(b)
The vector space $\mathfrak {g}\mathfrak {l}(n,\mathbb {R})$ of all $n\times n$ real matrices is an -dimensional Lie algebra under the commutator bracket . The canonical identification of both $\mathfrak {g}\mathfrak {l}(n,\mathbb {R})$ and $\mathrm{Lie}({{\,\mathrm{GL}\,}}(n,\mathbb {R}))$ with the tangent space to ${{\,\mathrm{GL}\,}}(n,\mathbb {R})$ at the identity yields a Lie algebra isomorphism $\mathfrak {g}\mathfrak {l}(n,\mathbb {R}) \cong \mathrm{Lie}({{\,\mathrm{GL}\,}}(n,\mathbb {R}))$ [ LeeSM , Prop. 8.41].
(c)
Similarly, the space $\mathfrak {g}\mathfrak {l}(n,\mathbb C)$ of $n\times n$ complex matrices is a -dimensional (real) Lie algebra under the commutator bracket, isomorphic to the Lie algebra of ${{\,\mathrm{GL}\,}}(n,\mathbb C)$ [ LeeSM , Prop. 8.48].
(d)
The space $\mathfrak o(n)$ of skew-symmetric $n\times n$ real matrices is a Lie subalgebra of $\mathfrak {g}\mathfrak {l}(n,\mathbb {R})$ , isomorphic to the Lie algebra of ${{\,\mathrm{O}\,}}(n)$ [ LeeSM , Example 8.47].
(e)
If V is a vector space, the space $\mathfrak {g}\mathfrak {l}(V)$ of all linear maps from V to itself is a Lie algebra under the commutator bracket $[A,B] = A\circ B - B\circ A$ . In case V is finite-dimensional, $\mathfrak {g}\mathfrak {l}(V)$ is isomorphic to the Lie algebra of the Lie group ${{\,\mathrm{GL}\,}}(V)$ .
(f)
A Lie algebra in which all brackets are zero is called an abelian Lie algebra . Every vector space can be made into an abelian Lie algebra by defining all brackets to be zero. The Lie algebra of a connected Lie group is abelian if and only if the group is abelian (see [ LeeSM , Problems 8-25 and 20-7]). //

The following proposition shows that every Lie group homomorphism induces a Lie algebra homomorphism between the respective Lie algebras.

Proposition C.5

(The Induced Lie Algebra Homomorphism). [ LeeSM , Thm. 8.44] Let G and H be Lie groups, and let $\mathfrak g$ and $\mathfrak h$ be their Lie algebras. Given a Lie group homomorphism $F:G\mathrel {\rightarrow }H$ , there is a unique Lie algebra homomorphism $F_*:\mathfrak g\mathrel {\rightarrow }\mathfrak h$ , called the induced Lie algebra homomorphism of ${\varvec{F}}$ , with the property that for each $X\in \mathfrak g$ , the vector field $F_*X\in \mathfrak h$ is F -related to X .

Proposition C.6

(Properties of Induced Homomorphisms). [ LeeSM , Prop. 8.45] Let G , H , K be Lie groups.

(a)
$(\mathrm{Id}_G)_* = \mathrm{Id}_{\mathrm{Lie}(G)}:\mathrm{Lie}(G)\mathrel {\rightarrow }\mathrm{Lie}(G)$ .
(b)
If $F_1:G\mathrel {\rightarrow }H$ and $F_2:H\mathrel {\rightarrow }K$ are Lie group homomorphisms, then
$\begin{aligned} (F_2\circ F_1)_* = (F_2)_*\circ (F_1)_*:\mathrm{Lie}(G)\mathrel {\rightarrow }\mathrm{Lie}(K). \end{aligned}$
(c)
Isomorphic Lie groups have isomorphic Lie algebras.

The Exponential Map of a Lie Group

If G is a Lie group, a Lie group homomorphism from $\mathbb {R}$ (with its additive group structure) to G is called a one-parameter subgroup of ${\varvec{G}}$ . (Note that the term “one-parameter subgroup” refers, confusingly, to a homomorphism, not to a Lie subgroup of G . But the image of a one-parameter subgroup is always a Lie subgroup of dimension at most 1.) Theorem 20.1 of [ LeeSM ] shows that for every $X\in \mathrm{Lie}(G)$ , the integral curve of X starting at the identity is a one-parameter subgroup, called the one-parameter subgroup generated by ${\varvec{X}}$ , and every one-parameter subgroup is of this form.

The exponential map of ${\varvec{G}}$ is the map $\exp ^G:\mathrm{Lie}(G)\mathrel {\rightarrow }G$ defined by setting $\exp ^G(X) = \gamma (1)$ , where $\gamma$ is the integral curve of X starting at e . (The exponential map is more commonly denoted simply by $\exp$ , but we use the notation $\exp ^G$ to distinguish the Lie group exponential map from the Riemannian exponential map, introduced in Chapter 5 .)

Proposition C.7

(Properties of the Lie Group Exponential Map). [ LeeSM , Props. 20.5 & 20.8] Let G be a Lie group and let $\mathfrak g$ be its Lie algebra.

(a)
$\exp ^G:\mathfrak g\mathrel {\rightarrow }G$ is smooth.
(b)
For every $X\in \mathfrak g$ , the map $\gamma :\mathbb {R}\mathrel {\rightarrow }G$ defined by $\gamma (t) = \exp ^G(tX)$ is the one-parameter subgroup generated by X .
(c)
The differential $\big (d\,\exp ^G\big )_0:T_0\mathfrak g \mathrel {\rightarrow }T_eG$ is the identity map, under the canonical identifications of $T_0\mathfrak g$ and with $\mathfrak g$ .

The exponential map is the key ingredient in the proof of the following fundamental result.

Theorem C.8

(Closed Subgroup Theorem). [ LeeSM , Thm. 20.12 & Cor. 20.13] Suppose G is a Lie group and $H\subseteq G$ is a subgroup in the algebraic sense. Then H is an embedded Lie subgroup of G if and only if it is closed in the topological sense.

An important special case of the closed subgroup theorem is that of a discrete subgroup , that is, a subgroup that is discrete in the subspace topology.

Proposition C.9.

[ LeeSM , Prop. 21.28] Every discrete subgroup of a Lie group is a closed Lie subgroup of dimension zero.

Adjoint Representations

Let G be a Lie group and $\mathfrak g$ its Lie algebra. For every $\varphi \in G$ , conjugation by $\varphi$ gives a Lie group automorphism $C_\varphi :G\mathrel {\rightarrow }G$ , called an inner automorphism , by $C_\varphi (\psi ) = \varphi \psi \varphi ^{-1}$ . Let ${{\,\mathrm{Ad}\,}}(\varphi ) = (C_\varphi )_*:\mathfrak g\mathrel {\rightarrow }\mathfrak g$ be the induced Lie algebra automorphism. It follows from the definition that $C_{\varphi _1}\circ C_{\varphi _2} = C_{\varphi _1\varphi _2}$ , and therefore ${{\,\mathrm{Ad}\,}}(\varphi _1)\circ {{\,\mathrm{Ad}\,}}(\varphi _2) = {{\,\mathrm{Ad}\,}}(\varphi _1 \varphi _2)$ ; in other words, ${{\,\mathrm{Ad}\,}}:G\mathrel {\rightarrow }{{\,\mathrm{GL}\,}}(\mathfrak g)$ is a representation, called the adjoint representation of ${\varvec{G}}$ . Proposition 20.24 in [ LeeSM ] shows that ${{\,\mathrm{Ad}\,}}$ is a smooth map.

Example C.10

(Adjoint Representations of Lie Groups).

(a)
If G is an abelian Lie group, then $C_\varphi$ is the identity map for every $\varphi \in G$ , so the adjoint representation is trivial: ${{\,\mathrm{Ad}\,}}(\varphi )=\mathrm{Id}_{\mathfrak g}$ for all $\varphi \in G$ .
(b)
Suppose G is a Lie subgroup of ${{\,\mathrm{GL}\,}}(n,\mathbb {R})$ . Then for each $A\in G$ , the conjugation map $C_A(B) = ABA^{-1}$ is the restriction to G of a linear map from the space $\mathrm{M}(n,\mathbb {R})$ of $n\times n$ matrices to itself. Its differential, therefore, is given by the same formula: ${{\,\mathrm{Ad}\,}}(A)X = AXA^{-1}$ for every $A\in G$ and $X\in \mathrm{Lie}(G)\subseteq \mathfrak {g}\mathfrak {l}(n,\mathbb {R})$ . //

There is also an adjoint representation for Lie algebras. If $\mathfrak g$ is any Lie algebra, a representation of $\varvec{\mathfrak {g}}$ is a Lie algebra homomorphism from $\mathfrak g$ to $\mathfrak {g}\mathfrak {l}(V)$ , the Lie algebra of all linear endomorphisms of some vector space V . For each $X\in \mathfrak g$ , define a map ${{\,\mathrm{ad}\,}}(X):\mathfrak g\mathrel {\rightarrow }\mathfrak g$ by ${{\,\mathrm{ad}\,}}(X)Y = [X,Y]$ . This defines ${{\,\mathrm{ad}\,}}$ as a map from $\mathfrak g$ to $\mathfrak {g}\mathfrak {l}(\mathfrak g)$ , and a straightforward computation shows that it is a representation of $\mathfrak g$ , called the adjoint representation of $\varvec{\mathfrak {g}}$ .

The next proposition shows how the two adjoint representations are related.

Proposition C.11.

[ LeeSM , Thm. 20.27] Let G be a Lie group and $\mathfrak g$ its Lie algebra, and let ${{\,\mathrm{Ad}\,}}:G\mathrel {\rightarrow }{{\,\mathrm{GL}\,}}(\mathfrak g)$ and ${{\,\mathrm{ad}\,}}:\mathfrak g\mathrel {\rightarrow }\mathfrak {g}\mathfrak {l}(\mathfrak g)$ be their respective adjoint representations. The induced Lie algebra representation ${{\,\mathrm{Ad}\,}}_*:\mathfrak g\mathrel {\rightarrow }\mathfrak {g}\mathfrak {l}(\mathfrak g)$ is given by ${{\,\mathrm{Ad}\,}}_* = {{\,\mathrm{ad}\,}}$ .

Group Actions on Manifolds

The most important applications of Lie groups in differential geometry involve their actions on other manifolds.

First we consider actions by abstract groups, not necessarily endowed with a smooth manifold structure or even a topology. Let G be a group and M a set. A left action of ${\varvec{G}}$ on ${\varvec{M}}$ is a map $G\times M\mathrel {\rightarrow }M$ , usually written $(\varphi , p) \mapsto \varphi \cdot p$ , satisfying $\varphi _1 \cdot (\varphi _2 \cdot p) = (\varphi _1\varphi _2) \cdot p$ and $e \cdot p = p$ for all $\varphi _1,\varphi _2\in G$ and $p\in M$ , where e is the identity element of G . Similarly, a right action of ${\varvec{G}}$ on ${\varvec{M}}$ is a map $M\times G\mathrel {\rightarrow }M$ satisfying $(p \cdot \varphi _1) \cdot \varphi _2 = p \cdot (\varphi _1\varphi _2)$ and $p \cdot e=p$ . In some cases, it will be important to give a name to an action such as $\theta :G\times M\mathrel {\rightarrow }M$ , in which case we write $\theta _\varphi (p)$ in place of $\varphi \cdot p$ , with a similar convention for right actions. Since right actions can be converted to left actions and vice versa by setting $\varphi \cdot p = p \cdot \varphi ^{-1}$ , for most purposes we lose no real generality by restricting attention to left actions. (But there are also situations in which right actions arise naturally.)

Given an action of G on M , for each $p\in M$ the isotropy subgroup at ${\varvec{p}}$ is the subgroup $G_p \subseteq G$ consisting of all elements that fix p : that is, $G_p= \{\varphi \in G: \varphi \cdot p = p\}$ . The group action is said to be free if $\varphi \cdot p=p$ for some $\varphi \in G$ and $p\in M$ implies $\varphi =e$ , or in other words, if $G_p = \{e\}$ for every p . It is said to be effective if $\varphi _1 \cdot p=\varphi _2 \cdot p$ for all p if and only if $\varphi _1=\varphi _2$ , or equivalently, if the only element of G that fixes every element of M is the identity. For effective actions, each element of G is uniquely determined by the map $p\mapsto \varphi \cdot p$ . In such cases, we will sometimes use the same notation $\varphi$ to denote either the element $\varphi \in G$ or the map $p\mapsto \varphi \cdot p$ . The action is said to be transitive if for every pair of points $p,q\in M$ , there exists $\varphi \in G$ such that $\varphi \cdot p = q$ .

If M is a smooth manifold, an action by a group G on M is said to be an action by diffeomorphisms if for each $\varphi \in G$ , the map $p\mapsto \varphi \cdot p$ is a diffeomorphism of M . If G is a Lie group, an action of G on a smooth manifold M is said to be a smooth action if the defining map $G\times M\mathrel {\rightarrow }M$ is smooth. In this case, it is also an action by diffeomorphisms, because each map $p\mapsto \varphi \cdot p$ is smooth and has $p\mapsto \varphi ^{-1} \cdot p$ as an inverse. If G is any countable group, an action by G on M is an action by diffeomorphisms if and only if it is a smooth action when G is regarded as a 0-dimensional Lie group with the discrete topology.

Example C.12

(Semidirect Products). Group actions provide an important way to construct Lie groups out of other Lie groups. Suppose H and N are Lie groups and $\theta :H\times N\mathrel {\rightarrow }N$ is a smooth left action of H on N by automorphisms of N , meaning that for each $h\in H$ , the map $\theta _h:N\mathrel {\rightarrow }N$ given by $\theta _h(n) = h \cdot n$ is a Lie group automorphism. The semidirect product of ${\varvec{H}}$ and ${\varvec{N}}$ determined by $\theta$ , denoted by ${N\mathbin {\rtimes }_\theta H}$ , is the Lie group whose underlying manifold is the Cartesian product $N\times H$ , and whose group multiplication is $(n,h)(n',h') = \big (n\theta _h(n'),hh'\big )$ . //

$\blacktriangleright$ Exercise C.13. Verify that ${N\mathbin {\rtimes }_\theta H}$ is indeed a Lie group with the multiplication defined above, and with ( e , e ) as identity and $(n,h)^{-1} = \big (\theta _{h^{-1}}(n^{-1}), h^{-1}\big )$ .

Now suppose G is a Lie group, and M and N are smooth manifolds endowed with G -actions (on the left, say). A map $F:M\mathrel {\rightarrow }N$ is said to be equivariant with respect to the given G actions if $F(\varphi \cdot x) = \varphi \cdot F(x)$ for all $\varphi \in G$ and $x\in M$ .

Theorem C.14

(Equivariant Rank Theorem). [ LeeSM , Thm. 7.25] Suppose M , N are smooth manifolds, and G is a Lie group acting smoothly and transitively on M and smoothly on N . If $F:M\mathrel {\rightarrow }N$ is a smooth G -equivariant map, then F has constant rank. Thus, if F is surjective, it is a smooth submersion; if it is injective, it is a smooth immersion; and if it is bijective, it is a diffeomorphism.

A smooth action of G on M is said to be a proper action if the map $G\times M\mathrel {\rightarrow }M\times M$ defined by $(\varphi ,x)\mapsto (\varphi \cdot x,x)$ is a proper map, meaning that the preimage of every compact set is compact. The following characterization is usually the easiest way to prove that a given action is proper.

Proposition C.15.

[ LeeSM , Prop. 21.5] Suppose G is a Lie group acting smoothly on a smooth manifold M . The action is proper if and only if the following condition is satisfied: if $$(p_i)$$ is a sequence in M and $(\varphi _i)$ is a sequence in G such that both and $(\varphi _i \cdot p_i)$ converge, then a subsequence of $(\varphi _i)$ converges.

Corollary C.16.

Every smooth action by a compact Lie group on a smooth manifold is proper.

Proof.

If G is a compact Lie group, then every sequence in G has a convergent subsequence, so every smooth G -action is proper by the preceding proposition. $\square$

The next theorem is the most important application of proper actions. If a group G acts on a manifold M , then each $p\in M$ determines a subset $G \cdot p = \{\varphi \cdot p: \varphi \in G\}$ , called the orbit of ${\varvec{p}}$ . Because two orbits are either identical or disjoint, the orbits form a partition of G . The set of orbits is denoted by M / G , and with the quotient topology it is called the orbit space of the action.

Theorem C.17

(Quotient Manifold Theorem). [ LeeSM , Thm. 21.10] Suppose G is a Lie group acting smoothly, freely, and properly on a smooth manifold M . Then the orbit space M / G is a topological manifold whose dimension is equal to the difference $\dim M - \dim G,$ and it has a unique smooth structure with the property that the quotient map $\pi :M\mathrel {\rightarrow }M/G$ is a smooth submersion.

Example C.18

(Real Projective Spaces). For each nonnegative integer n , the n - dimensional real projective space , denoted by $\mathbb {R}\mathbb P^n$ , is defined as the set of one-dimensional linear subspaces of $\mathbb {R}^{n+1}$ . It can be identified with the orbit space of ../images/56724_2_En_BookBackmatter_IEq1429_HTML.gif under the action of the group $\mathbb {R}^*$ of nonzero real numbers given by scalar multiplication: $\lambda \cdot \big (x^1,\dots ,x^{n+1}\big ) = \big (\lambda x^1,\dots , \lambda x^{n+1}\big )$ . It is easy to check that this action is smooth and free. To see that it is proper, we use Proposition C.15 . Suppose $$(x_i)$$ is a sequence in ../images/56724_2_En_BookBackmatter_IEq1433_HTML.gif and $(\lambda _i)$ is a sequence in $\mathbb {R}^*$ such that and . Then $|\lambda _i | = |\lambda _ix_i|/|x_i|$ converges to the nonzero real number | y | / | x |. Thus the numbers $\lambda _i$ all lie in a compact set of the form $\{\lambda : 1/C \le |\lambda | \le C\}$ for some positive number C , so a subsequence converges to a nonzero real number. Therefore, by the quotient manifold theorem, $\mathbb {R}\mathbb P^n$ has a unique structure as a smooth n -dimensional manifold such that the quotient map ../images/56724_2_En_BookBackmatter_IEq1442_HTML.gif is a smooth submersion. //

Example C.19

(Complex Projective Spaces). Similarly, the n - dimensional complex projective space , denoted by $\mathbb C\mathbb P^n$ , is the set of 1-dimensional complex subspaces of $\mathbb C^{n+1}$ , identified with the orbit space of ../images/56724_2_En_BookBackmatter_IEq1446_HTML.gif under the $\mathbb C^*$ -action given by $\lambda \cdot z = \lambda z$ . The same argument as in the preceding example shows that this action is smooth, free, and proper, so $\mathbb C\mathbb P^n$ is a smooth 2 n -dimensional manifold and the quotient map ../images/56724_2_En_BookBackmatter_IEq1450_HTML.gif is a smooth submersion. //

Group Actions and Covering Spaces

There is a close connection between smooth covering maps and smooth group actions. To begin, suppose ${{\widetilde{M}}}$ and M are smooth manifolds and $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a smooth covering map. A covering automorphism of ${\varvec{\pi }}$ is a diffeomorphism $\varphi :{{\widetilde{M}}}\mathrel {\rightarrow }{{\widetilde{M}}}$ such that $\pi \circ \varphi =\pi$ ; the set of all covering automorphisms is a group under composition, called the covering automorphism group and denoted by $\mathrm{Aut}_{\pi }\big ({{\widetilde{M}}}\big )$ .

Proposition C.20.

[ LeeSM , Prop. 21.12] Let $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ be a smooth covering map. With the discrete topology, $\mathrm{Aut}_{\pi }\big ({{\widetilde{M}}}\big )$ is a discrete Lie group acting smoothly, freely, and properly on ${{\widetilde{M}}}$ .

Because of the requirement that a covering automorphism $\varphi$ satisfy $\pi \circ \varphi = \pi$ , it follows that $\varphi$ restricts to an action on each fiber of $\pi$ . The next proposition describes the conditions in which this action is transitive. A covering map $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is called a normal covering if the image of the homomorphism $\pi _*:\pi _1\big ({{\widetilde{M}}},{\tilde{x}}\big )\mathrel {\rightarrow }\pi _1\big (M,\pi ({\tilde{x}})\big )$ is a normal subgroup of $\pi _1\big (M,\pi ({\tilde{x}})\big )$ . (Note that every universal covering is a normal covering, because the trivial subgroup is always normal.)

Proposition C.21.

[ LeeTM , Cor. 12.5] If $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is a smooth covering map, then the automorphism group of $\pi$ acts transitively on each fiber of $\pi$ if and only if $\pi$ is a normal covering.

The universal covering is the most important special case.

Proposition C.22

(Automorphisms of the Universal Covering). [ LeeTM , Cor. 12.9] Suppose M is a connected smooth manifold and $\pi :{{\widetilde{M}}}\mathrel {\rightarrow }M$ is its universal covering. Then $\mathrm{Aut}_{\pi }\big ({{\widetilde{M}}}\big )$ is isomorphic to the fundamental group of M , and it acts transitively on each fiber of $\pi$ .

The next theorem, which is an application of the quotient manifold theorem, is a partial converse to Proposition C.20 .

Proposition C.23.

[ LeeSM , Thm. 21.13] Suppose ${{\widetilde{M}}}$ is a connected smooth manifold and $\varGamma$ is a discrete Lie group acting smoothly, freely, and properly on ${{\widetilde{M}}}$ . Then the orbit space ${{\widetilde{M}}}/\varGamma$ has a unique smooth manifold structure such that the quotient map ${{\widetilde{M}}}\mathrel {\rightarrow }{{\widetilde{M}}}/\varGamma$ is a smooth normal covering map.

Example C.24

(The Universal Covering of ${{\varvec{\mathbb {R}}}}{\varvec{\mathbb {P}}}^{{\varvec{n}}}$ ). The two-element group $\varGamma = \{\pm 1\}$ acts smoothly, freely, and properly on $\mathbb {S}^n$ by multiplication, and thus $\mathbb {S}^n/\varGamma$ is a smooth manifold and the quotient map $\pi :\mathbb {S}^n \mathrel {\rightarrow }\mathbb {S}^n/\varGamma$ is a two-sheeted smooth normal covering map. Note also that the quotient map ../images/56724_2_En_BookBackmatter_IEq1485_HTML.gif defined in Example C.18 restricts to a surjective smooth map $q:\mathbb {S}^{n}\mathrel {\rightarrow }\mathbb {R}\mathbb P^n$ , which is a submersion because each point of $\mathbb {S}^n$ is in the image of a smooth local section. Since q and $\pi$ are constant on each other’s fibers, Proposition A.19 shows that there is a diffeomorphism $\mathbb {S}^n / \varGamma \mathrel {\rightarrow }\mathbb {R}\mathbb P^n$ , and q is equal to the composition $\mathbb {S}^n \mathrel {\rightarrow }\mathbb {S}^n / \varGamma \mathrel {\rightarrow }\mathbb {R}\mathbb P^n$ . Since this is a smooth normal covering map followed by a diffeomorphism, q is also a smooth normal covering map. Thus $\mathbb {S}^n$ is the universal covering space of $\mathbb {R}\mathbb P^n$ , and the fundamental group of $\mathbb {R}\mathbb P^n$ is isomorphic to $\{\pm 1\}$ . //

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Notational Index

Symbols

$\wedge$ (wedge product)

$\bigcirc\!\!\!\!\!\!\!\wedge$ (Kulkarni–Nomizu product)

$\amalg$ (disjoint union)

$\lrcorner$ (interior multiplication)

$\sharp$ (sharp)

$\ast$ (Hodge star operator)

$\vert \centerdot \vert$ (length of a vector)

$\vert \centerdot \vert _g$ (length of a vector)

$[\mathbin {bad hbox} ]$ (path class)

$[\centerdot ,\centerdot ]$ (commutator bracket)

$[\mathbin {bad hbox} ,\mathbin {bad hbox} ]$ (Lie bracket)

$(\centerdot ,\centerdot )$ (global inner product on forms)

$\langle \centerdot ,\centerdot \rangle$ (inner product)

$\langle \centerdot ,\centerdot \rangle$ (pairing between vector and covector)

$\langle \centerdot ,\centerdot \rangle$ (scalar product)

$\langle \centerdot ,\centerdot \rangle _g$ (inner product)

A

(adjoint of a matrix)

(transpose of a matrix)

$\mathscr {A}(TM)$ (set of connections on $$TM $$

)

${\mathrm{ad}}$ (adjoint representation of a Lie algebra)

${\mathrm{Ad}}$ (adjoint representation of a Lie group)

${\mathrm{Alt}}$ (alternating projection)

${\mathrm{Aut}}_{\pi }\big (\big )$ (covering automorphism group)

B

$\flat$ (flat)

(geodesic ball)

$\bar{B}_c(p)$ (closed geodesic ball)

$\mathbb B ^n(R)$ (Poincaré ball)

C

(Cotton tensor)

$C^{\infty }$ (infinitely differentiable)

$C^{\infty }(M)$ (smooth real-valued functions)

$C^{\infty }(M,N)$ (smooth maps)

${\mathbb {CP}}^n$ (complex projective space)

$\mathrm{{conv}}(x)$ (convexity radius at $$x $$

)

$\mathrm{{Cut}}(p)$ (cut locus of $$p $$

)

D

$\delta ^i_j$ (Kronecker delta)

$\Delta$ (Laplacian)

$\partial /\partial x^i$ (coordinate vector field)

$\partial /\partial x^i\vert _p$ (coordinate vector)

$\partial _i$ (coordinate vector field)

$\partial _i\vert _p$ (coordinate vector)

$\partial_r$ (radial vector field)

$\partial_s\Gamma$ (velocity of a transverse curve)

$\partial_t\Gamma$ (velocity of a main curve)

$\Delta$ (covariant derivative)

$\Delta^{\perp }$ (normal connection)

$\nabla^{\top }$ (tangential connection)

$\nabla^2u$ (covariant Hessian)

$\nabla ^2_{X,Y}$ (second total covariant derivative)

$\nabla F$ (total covariant derivative)

$\nabla_XY$ (covariant derivative)

(exterior derivative)

(exterior covariant derivative)

(adjoint of $$d$$

)

(differential of a map)

(Riemannian distance)

(distance to a subset)

$\mathscr{D} M$ (density bundle)

$\mathscr{D}^{(p)}$ (domain of $\theta ^{(p)}$ )

(covariant derivative along transverse curves)

(covariant derivative along a curve)

(Riemannian density)

(Riemannian volume form)

div

(divergence of $$X $$

)

E

(identity of a Lie group)

(normal exponential map)

$\mathscr{E}$ (domain of the exponential map)

$E\vert _M$ (restriction of a bundle)

(Euclidean group)

$\mathscr{E}_P$ (domain of the normal exponential map)

$\mathscr{E}_p$ (domain of the restricted exponential map)

$\mathrm{End}(V)$ (space of endomorphisms)

$\exp$ (exponential map)

$\exp ^G$ (exponential map of a Lie group)

$\exp _p$ (restricted exponential map)

F

$\varphi ^*{\nabla }$ (pullback of a connection)

$\Phi ^{-1}(\{y\})$ (level set)

$\Phi ^{-1}(y)$ (level set)

(induced fundamental group homomorphism)

(induced Lie algebra homomorphism)

(pushforward of vector fields)

(pullback of a density)

(pullback of a tensor field)

(symmetric product of $$F $$

and

)

G

${\gamma ^{\prime }}$ (velocity vector)

$\gamma ^{\prime }\big (a_i^{\pm }\big )$ (one-sided velocity vectors)

$\gamma _v$ (geodesic with initial velocity $$v $$

)

$\varGamma (E)$ (smooth sections of $$ E $$

)

$\varGamma _{ij}^k$ (connection coefficients)

$\varGamma (s,t)$ (family of curves)

$\hat{g}$ (pullback of the round metric)

${\bar{g}}$ (Euclidean metric)

(round metric)

${mathring}256g_R$ (round metric of radius R)

$\breve{g}$ (hyperbolic metric)

$\breve{g}_R$ (hyperbolic metric of radius $$R $$

)

$g_{ij}$ (metric coefficients)

$g^{ij}$ (inverse of $g_{ij}$ )

GL $(n,\mathbb{C})$ (complex general linear group)

GL $(n,\mathbb{R})$ (general linear group)

${\mathfrak {gl}}(n,\mathbb{R})$ (matrix Lie algebra)

${\mathfrak {gl}}(V)$ (Lie algebra of linear maps)

(group generated by commutators)

$g_1 \oplus f^2 g_2$ (warped product metric)

$g_1\oplus g_2$ (product metric)

(group of invertible linear maps)

(gradient of $$f $$

)

H

(scalar second fundamental form)

(mean curvature)

${\mathbb{H}}n$ (hyperbolic space)

${{\mathbb{H}}^n}(R)$ (hyperbolic space of radius $$R $$

)

(Heisenberg group)

$\mathring{\mathscr {H}}_r$ (traceless Hessian operator)

$\mathscr {H}_r$ (Hessian operator)

${{\mathbb{H}}^{r,s}}(R)$ (pseudohyperbolic space)

Hol

(holonomy group)

Hol

(restricted holonomy group)

I

(interior multiplication)

(identity matrix)

(index form)

$\mathrm{{ID}}(p)$ (injectivity domain)

$\mathrm {II}$ (second fundamental form)

$\mathrm {II}_N$ (second fundamental form in direction $$N $$

)

$\mathrm{{inj}}(M)$ (injectivity radius of $$M $$

)

$\mathrm{{inj}}(p)$ (injectivity radius at $$p $$

)

${{\mathrm{Iso}}}(M,g)$ (isometry group)

${{\mathrm{Iso}}}_p(M,g)$ (isotropy group of a point)

J

$J(\gamma )$ (space of Jacobi fields)

$\mathscr {J}^\perp (\gamma )$ (normal Jacobi fields)

$\mathscr {J}^\top (\gamma )$ (tangential Jacobi fields)

K

$\kappa$ (Cayley transform)

$\kappa$ (geodesic curvature)

$\kappa _i$ (principal curvatures)

$\kappa _N$ (signed curvature)

(Gaussian curvature)

$\mathbb K ^n(R)$ (Beltrami–Klein model)

L

$\Lambda^kT^*M$ (bundle of alternating tensors)

$\Lambda^k(V^*)$ (space of alternating tensors)

$L_g(\gamma )$ (length of curve)

$\mathrm{Lie}(G)$ (Lie algebra of left-invariant vector fields on $$G $$

)

M

$\vert \mu \vert$ (density associated with an $$n $$

-form)

(

with opposite orientation)

(orbit space)

$(n\times k,\mathbb R )$ (space of matrices)

$\mathrm{M}(n,\mathbb R )$ (space of $n\times n$ matrices)

N

(normal space)

(normal bundle)

O

$\omega _i{}^j$ (connection $$1 $$

-forms)

$\Omega ^k(M)$ (space of $$k $$

-forms)

${{\mathrm{O}}}(M)$ (orthonormal bases on $$M $$

)

${{\mathrm{O}}}(n)$ (orthogonal group)

${{\mathrm{O}}}(n,1)$ (Lorentz group)

${{\mathrm{O}}}^+(n,1)$ (orthochronous Lorentz group)

${{\mathrm{O}}}(n+1)$ (orthogonal group)

P

$\pi _1(M, p)$ (fundamental group)

$\pi ^{\perp }$ (normal projection)

$\pi ^{\top }$ (tangential projection)

(Schouten tensor)

$P^\gamma _{t_0t_1}$ (parallel transport operator)

Q

$bar{q}^{(r,s)}$ (pseudo-Euclidean metric)

${breve}24Dq^{(r,s)}_R$ (pseudohyperbolic metric)

${mathring}256q^{(r,s)}_R$ (pseudospherical metric)

R

$\rho (\gamma )$ (rotation index)

(radial distance function)

(curvature endomorphism)

$\mathbb{R}^n$ (Euclidean space)

$\mathbb{R}^{r,s}$ (pseudo-Euclidean space)

$\mathbb {RP} ^n$ (real projective space)

${mathscr{R}}(V^*)$ (space of algebraic curvature tensors)

(curvature endomorphism)

(curvature endomorphism)

(Ricci tensor)

(curvature tensor)

S

$\Sigma ^k T^*M$ (bundle of symmetric tensors)

$\Sigma^k(V^*)$ (space of symmetric tensors)

(shape operator)

(scalar curvature)

(surface of revolution)

(geodesic sphere)

$\mathbb S ^n$ (unit $$n $$

-sphere)

$\mathbb S ^n(R)$ ( $$n $$

-sphere of radius $$R $$

)

$S^{\perp }$ (orthogonal complement)

§ $mathbb{S}^{r,s}(R)$ (pseudosphere)

$\mathrm{SL}(n,\mathbb C )$ (complex special linear group)

$\mathrm{SL}(n,\mathbb R )$ (special linear group)

$\mathrm{SO}(n)$ (special orthogonal group)

$\mathrm{SU}(n)$ (special unitary group)

$\sec (\Pi )$ (sectional curvature)

$\sec (v,w)$ (sectional curvature)

$\mathrm{sgn}$ (sign of a permutation)

$\mathrm Sol$ ( $$3 $$

-dimensional solvable Lie group)

$\mathrm{supp}\,f$ (support of $$f $$

)

${\mathrm{Sym}}$ (symmetrization)

T

$\tau$ (torsion tensor)

$\mathbb{T}n$ ( $$n $$

-torus)

(tangent bundle)

$T\vert _{M}$ (ambient tangent bundle)

(tangent space)

$T^\top _{\gamma (t)}M$ (space of vectors tangent to a curve)

$T^\perp _{\gamma (t)}M$ (space of vectors normal to a curve)

$\mathscr{T}^k(M)$ (space of tensor fields)

(bundle of contravariant $$k $$

-tensors)

(bundle of covariant $$k $$

-tensors)

(space of contravariant $$k $$

-tensors)

(space of covariant $$k $$

-tensors)

$T^{(k,l)}TM$ (bundle of mixed tensors)

$T^{(k,l)}(V)$ (space of mixed tensors)

$\mathrm{{TCL}}(p)$ (tangent cut locus)

$\mathrm{tr}$ (trace of a tensor)

$\mathrm{tr} _g$ (trace with respect to $$g $$

)

U

$\mathrm{U}(n)$ (unitary group)

$\mathbb U ^n$ (Poincaré half-space)

V

(dual space of $$V $$

)

$\vert v\wedge w\vert$ (norm of an alternating $$2 $$

-tensor)

$\mathrm V _k(\mathbb R ^n)$ (Stiefel manifold)

(volume of a Riemannian manifold)

W

(Weyl tensor)

(Weingarten map in direction $$N $$

)

X

$\mathbin {\rtimes }$ (semidirect product)

$\chi (M)$ (Euler characteristic)

$\mathfrak X (M)$ (space of vector fields)

$X^{\perp }$ (normal projection of $$X $$

)

$\mathfrak X ^\perp (\gamma )$ (normal vector fields along a curve)

$X^{\top }$ (tangential projection of $$X $$

)

$\mathfrak X ^\top (\gamma )$ (tangential vector fields along a curve)

$\mathfrak X(\gamma )$ (space of vector fields along a curve)

Subject Index

Symbols

-Hessian

-center

-point homogeneous

-point homogeneous

A

abelian Lie algebra

absolute derivative

acceleration

in $\mathbb R ^n$

of a curve in a manifold

of a plane curve

tangential

action, see group action

adapted frame

adjoint matrix

adjoint representation

of a Lie algebra

of a Lie group

admissible curve

admissible family

admissible loop

admissible partition

for a curve

for a family of curves

for a vector field along a curve

affine connection

aims at a point

algebraic Bianchi identity

algebraic curvature tensor

Alt convention for wedge product

alternating tensor

alternation

ambient manifold

ambient tangent bundle

Ambrose, Warren

analytic continuation of a local isometry

angle

between vectors

of a geodesic triangle

tangent

angle excess

angle-sum theorem

anti-de Sitter space

universal

anti-self-dual

anticommutative

antipodal points

arc length, parametrization by

arc-length function

area-minimizing hypersurface

area of a hypersurface

aspherical

asymptotically parallel

atlas

automorphism of a Lie group

inner

Avez, André

axial isometry

axis for an isometry

B

Böhm, Christoph

Böhm–Wilking theorem

backward reparametrization

ball

geodesic

metric

Poincaré

regular coordinate

smooth coordinate

volume of

base of a vector bundle

basis isomorphism

Beltrami–Klein model

Berger sphere

Berger, Marcel

bi-invariant metric

curvature of

existence of

exponential map of

Bianchi identity

algebraic

contracted

differential

first

second

Bieberbach, Ludwig

bilinear form

Bishop, Richard L.

Bishop–Gromov theorem

Bochner's formula

Bonnet, Ossian

boundary chart

boundary normal coordinates

boundary of a manifold with boundary

boundary problem, two-point

boundary slice coordinates

bounded subset

bracket

commutator

in a Lie algebra

Brendle, Simon

bump function

bundle

normal

of tensors

vector

C

CR manifold

Calabi–Yau manifold

calculus of variations

canonical form

for a nonvanishing vector field

for commuting vector fields

Carathéodory metric

Carnot–Carathéodory metric

Cartan's first structure equation

Cartan's fixed-point theorem

Cartan's second structure equation

Cartan's torsion theorem

Cartan, Élie

Cartan–Ambrose–Hicks theorem

Cartan–Hadamard manifold

Cartan–Hadamard theorem

catenoid

Cauchy–Riemann manifold

Cayley transform

central projection

chart

containing a point

smooth

Cheeger, Jeff

Cheng's maximal diameter theorem

Cheng, Shiu-Yuen

Chern, Shiing-Shen

Chern–Gauss–Bonnet theorem

Choquet-Bruhat, Yvonne

Christoffel symbols

circle classification theorem

circle group

circumference theorem

Clarke, Chris J. S.

classification theorem

circle

compact surfaces

constant-curvature metrics

plane curve

closed form

closed geodesic

closed geodesic ball

closed map

closed subgroup theorem

Codazzi equation

for a hypersurface

Codazzi tensor

codimension

coframe

commutator bracket

commuting vector fields

canonical form

comparison theorem

Bishop–Gromov

conjugate point

diameter

Günther

Hessian

injectivity radius

Jacobi field

Laplacian

metric

principal curvature

Riccati

volume

compatibility with a metric

complete Riemannian manifold

geodesically

metrically

complex projective space

component functions

composable paths

conformal diffeomorphism

conformal invariance

of the Cotton tensor

of the Weyl tensor

conformal Laplacian

conformal metrics

conformal transformation

of the curvature

of the Levi-Civita connection

conformally equivalent metrics

conformally flat, locally

hyperbolic space

spheres

conformally related metrics

congruent

conjugate point

critical point of $\exp _p$

geodesic not minimizing past

conjugate point comparison theorem

connection

Euclidean

existence of

flat

in a vector bundle

in components

in the tangent bundle

Koszul

naturality of

normal

on a manifold

on tensor bundles

Riemannian

tangential

connection $$1 $$

-forms

connection coefficients

transformation law for

constant Gaussian curvature

constant mean curvature

constant rank

constant sectional curvature

characterization of

classification

formula for curvature tensor

formula for metric

local uniqueness

model spaces

uniqueness

constant-speed curve

constraint equations, Einstein

contracted Bianchi identity

contraction

contravariant tensor

control theory

convex hypersurface

convex subset

convex, geodesically

convexity radius

coordinate representation

coordinate ball

regular

smooth

coordinate chart

smooth

coordinate domain

coordinate frame

coordinate vector

coordinate vector field

coordinates

boundary slice

Fermi

graph

have upper indices

local

natural, on tangent bundle

normal

polar Fermi

polar normal

semigeodesic

slice

standard, on $\mathbb{R }^n$

cosmological constant

cotangent bundle

cotangent space

Cotton tensor

conformal invariance of

covariant derivative

along a curve

exterior

of tensor field

second

total

covariant Hessian

covariant tensor

covector

covector field

covering automorphism

covering automorphism group

covering map

homeomorphism criterion

normal

Riemannian

smooth

universal

critical point

of the exponential map

critical value

crystallographic groups

curl

curvature

conformal transformation of

constant sectional

Gaussian

geodesic

mean

of a curve in a manifold

of a plane curve

principal

Ricci

Riemann

scalar

sectional

signed

curvature

-forms

curvature endomorphism

curvature operator

curvature tensor

determined by sectional curvatures

is isometry invariant

symmetries of

curve

admissible

in a manifold

piecewise regular

plane

smooth

curve segment

curved polygon

curved triangle

cusp vertex

cut locus

and injectivity radius

tangent

cut point

cut time

is continuous

cylinder, principal curvatures

D

$\delta$ -pinched

pointwise

strictly

Darboux theorem

de Sitter space

defining function

local

del

delta, Kronecker

density

on a manifold

on a vector space

pullback of

Riemannian

density bundle

derivation

of $C^\infty (M)$

determinant convention for wedge product

diameter

diameter comparison theorem

diffeomorphism

in $\mathbb{R }^n$

local

difference tensor

differentiable sphere theorem

differential

global

of a function

of a map

differential Bianchi identity

differential form

closed

exact

dihedral groups

direct product of Lie groups

directional derivative of a vector field on $$^n $$

Dirichlet eigenvalue

Dirichlet's principle

discrete Lie group

discrete subgroup

disjoint union

distance function

has geodesic integral curves

has unit gradient

local

distance, Riemannian

on a disconnected manifold

diverge to infinity

divergence

in coordinates

in terms of covariant derivatives

of a tensor field

divergence theorem

domain of the exponential map

dot product

double elliptic geometry

double of a manifold

dual basis

dual coframe

dual space

E

$\varepsilon$ -tubular neighborhood

edge of a curved polygon

effective group action

eigenfunction of the Laplacian

eigenvalue of the Laplacian

Dirichlet

Neumann

Einstein constraint equations

Einstein equation

Einstein field equation

Einstein metric

-dimensional

Einstein summation convention

Einstein, Albert

Einstein–Hilbert action

embedded submanifold

embedding

isometric

endomorphism

curvature

equivalent to a $$(1,1) $$

tensor

of a vector space

energy functional

equivariant map

equivariant rank theorem

escape lemma

Euclidean connection

Euclidean geodesics

Euclidean group

Euclidean metric

Euclidean space

sectional curvature of

Euclidean space form

Euclidean triangle

Euclidean, locally

Euler characteristic

Euler–Lagrange equation

evenly covered neighborhood

exact form

existence and uniqueness

for linear ODEs

of geodesics

of Jacobi fields

exponential map

differential of

domain of

naturality of

normal

of a bi-invariant metric

of a Lie group

of a Riemannian manifold

extendible Riemannian manifold

extendible vector fields

extension

of functions

of sections of vector bundles

exterior angle

exterior covariant derivative

exterior derivative

naturality of

extrinsic curvature of a curve

F

-related

faithful representation

family of curves

admissible

Fermi coordinates

polar

Fermi, Enrico

fiber

of a map

of a vector bundle

fiber metric

on a tensor bundle

on differential forms

figure eight

Finsler metric

first Bianchi identity

first fundamental form

first structure equation

first variation

flat ( $\flat$ )

flat connection

flat metric

flat vertex

flatness criterion

flow

fundamental theorem on

flow domain

focal point

form

bilinear

closed

exact

forward reparametrization

frame

coordinate

global

local

frame-homogeneous

locally

Frankel, Theodore

free group action

free homotopy class

trivial

freely homotopic

Fubini–Study metric

curvature of

functional

length

linear

fundamental form

first

second

fundamental group

homotopy invariance

fundamental theorem

of hypersurface theory

of Riemannian geometry

on flows

G

Gauss equation

for a hypersurface

Gauss formula

along a curve

for a curve in a hypersurface

for a hypersurface

Gauss lemma

for submanifolds

Gauss map

Gauss's Theorema Egregium

Gauss, Carl Friedrich

Gauss–Bonnet formula

Gauss–Bonnet theorem

Gaussian curvature

constant

is isometry invariant

of an abstract $$2 $$

-manifold

general linear group

general relativity

generating curve

genus

geodesic

closed

existence and uniqueness of

is locally minimizing

maximal

naturality of

on Euclidean space

on hyperbolic spaces

on spheres

pseudo-Riemannian

radial

Riemannian

with a conjugate point

with respect to a connection

geodesic ball

closed

geodesic curvature

geodesic equation

geodesic loop

geodesic polygon

geodesic segment

in path-homotopy class

geodesic sphere

geodesic triangle

geodesic vector field

geodesically complete

equivalent to metrically complete

geodesically convex

geometrization conjecture

global differential

global frame

gradient

Gram–Schmidt algorithm

graph coordinates

graph of a smooth function

graph parametrization

Grassmann manifold

Grassmannian

Gray, Alfred

great circle

great hyperbola

Green's identities

Greene, Robert

Gromoll, Detlef

Gromov, Misha

group action

by diffeomorphisms

by isometries

effective

free

isometric

left

proper

right

smooth

transitive

Günther, Paul

Günther's volume comparison theorem

H

Hadamard, Jacques

half-cylinder, principal curvatures

half-space, Poincaré

Hamilton's $$3 $$

-manifold theorem

Hamilton's $$4 $$

-manifold theorem

Hamilton, Richard

harmonic function

Heisenberg group

Hermann, Robert

Hessian

covariant

of length functional

Hessian comparison theorem

Hessian operator

Hicks, Noel

Hilbert, David

Hodge star operator

holonomy

restricted

homogeneous

-point

-point

-point

locally

homogeneous Riemannian manifold

homogeneous space

homomorphism

Lie algebra

Lie group

homotopic maps

homotopic, freely

homotopy

homotopy equivalence

Hopf, Heinz

Hopf–Rinow theorem

horizontal index position

horizontal lift

horizontal tangent space

horizontal vector field

hyperbolic metric

hyperbolic plane

hyperbolic space

distance function of

locally conformally flat

sectional curvature of

hyperbolic space form

hyperbolic stereographic projection

hyperboloid model

hypersurface

I

ideal triangle

immersed submanifold

with boundary

immersion

isometric

index

of a geodesic segment

of a scalar product

upper and lower

index form

index positions

induced homomorphism

of fundamental groups

of Lie algebras

induced metric

on a submanifold

induced orientation

infinitesimal generator

injectivity domain

injectivity radius

and cut locus

continuity of

injectivity radius comparison theorem

injectivity theorem for covering maps

inner automorphism

inner product

of tensors

inner product space

integral

of a function

with respect to arc length

integral curve

integral of an $$n $$

-form

integration by parts

interior angle

interior chart

interior multiplication

interior of a curved polygon

interior of a manifold with boundary

interpretation of an axiomatic system

intrinsic curvature of a curve

intrinsic property

invariant inner product

invariant metric

invariant under a flow

invariant, local

inverse function theorem

inward-pointing normal

irreducible symmetric space

isometric embedding

isometric group action

isometric immersion

isometric manifolds

isometries

of Euclidean space

of hyperbolic spaces

of spheres

isometry

axial

linear

local

metric

of a Riemannian manifold

Riemannian

isometry group

of Euclidean space

of hyperbolic spaces

of spheres

isomorphism of Lie groups

isothermal coordinates

isotropic

at a point

isotropy representation

isotropy subgroup

J

Jacobi equation

Jacobi field

existence and uniqueness

normal

on constant-curvature manifolds

tangential

transverse

Jacobi field comparison theorem

Jacobi identity

Jacobian matrix

jumps in tangent angle

K

-point homogeneous

Kazdan, Jerry

Killing vector field

Killing–Hopf theorem

Klein bottle

Klingenberg, Wilhelm

Kobayashi metric

Koszul connection

Koszul's formula

Kronecker delta

Kulkarni–Nomizu product

L

Lagrange multiplier

Laplace–Beltrami operator

Laplacian

conformal

Dirichlet eigenvalue of

eigenfunction of

eigenvalue of

in coordinates

Neumann eigenvalue of

Laplacian comparison theorem

latitude circle

lattice

law of inertia, Sylvester's

left action

left-invariant metric

left-invariant vector field

left translation

length

additivity of

isometry invariance of

of a curve

of a group element

of a vector

parameter independence of

length functional

lens space

level set

Levi-Civita connection

naturality of

Levi-Civita, Tullio

Lichnerowicz's theorem

Lie algebra

abelian

of a Lie group

Lie algebra automorphism

Lie algebra homomorphism

induced

Lie algebra isomorphism

Lie bracket

naturality of

of vectors tangent to a submanifold

Lie derivative

and Lie bracket

of a tensor field

of a vector field

Lie group

direct product of

discrete

semidirect product of

Lie group homomorphism

Lie group isomorphism

Lie subalgebra

Lie subgroup

lift of a map

from simply connected spaces

lifting criterion for covering maps

lifting property

path

unique

line in a Riemannian manifold

linear connection

linear functional

linear isometry

linear ODEs

local coordinates

local defining function

local diffeomorphism

local distance function

local frame

local invariant

local isometry

local parametrization

local section

of a submersion

local-to-global theorem

local trivialization

local uniqueness of constant-curvature metrics

locally conformally flat

-manifolds

hyperbolic space

spheres

locally Euclidean

locally finite

locally frame-homogeneous

locally homogeneous

locally minimizing curve

locally symmetric space

and parallel curvature

loop

geodesic

Lorentz group

Lorentz metric

existence of

lowering an index

M

main curve

manifold

smooth

topological

with boundary

without boundary

matrix Lie algebra

matrix Riccati comparison theorem

maximal diameter theorem

maximal geodesic

mean curvature

meridian

metric

bi-invariant

Carathéodory

Carnot–Carathéodory

Euclidean

fiber

Finsler

Fubini–Study

hyperbolic

induced by an immersion

Kobayashi

Lorentz

on a tensor bundle

on differential forms

product

pseudo-Riemannian

Riemannian

round

semi-Riemannian

singular Riemannian

sub-Riemannian

metric ball

metric comparison theorem

metric connection

metric isometry

metric space

metrically complete

equivalent to geodesically complete

Milnor's theorem on growth of the fundamental group

Milnor, John

minimal hypersurface

minimal surface

minimizing curve

is a geodesic

locally

Minkowski metric

Minkowski space

mixed tensor

Möbius band

model of an axiomatic system

model Riemannian manifolds

monodromy theorem

for local isometries

for paths

Morse index theorem

multilinear

over $C^{\infty }(M)$

multiplicity of conjugacy

Munkres, James

musical isomorphisms

commute with covariant derivatives

Myers's theorem

Myers, Sumner B.

Myers–Steenrod theorem

N

-manifold

-torus

flat metric on

nabla

Nash embedding theorem

Nash, John

natural coordinates on the tangent bundle

naturality

of geodesics

of the exponential map

of the Levi-Civita connection

of the Lie bracket

negative definite

negatively oriented basis

Neumann eigenvalue

nilpotent Lie group

noncollinear

nondegenerate $$2 $$

-tensor

nondegenerate bilinear form

nondegenerate $$n $$

-tuple of vector fields

nondegenerate subspace

nonvanishing vector field

norm

on a vector space

on an inner product space

pseudo-Riemannian

normal bundle

pseudo-Riemannian

normal connection

normal coordinates

polar

pseudo-Riemannian

Riemannian

normal covering map

normal exponential map

normal Jacobi field

normal neighborhood

of a point

of a submanifold

normal projection

normal space

normal variation

normal vector

outward-pointing

normal vector field

along a curve

O

O'Neill's formula

one-parameter family of curves

one-parameter subgroup

one-sided velocity vectors

open geodesic ball

open map

open Möbius band

orbit

orbit space

order of conjugacy

ordered basis

ordinary differential equation

linear

ordinary vertex

orientable manifold

orientation

and coordinates

and

-forms

continuous

induced

of a boundary

of a curved polygon

of a hypersurface

of a manifold

of $\mathbb R ^n$

orientation covering

orientation form

orientation-preserving

orientation-reversing

oriented basis

oriented coordinate chart

oriented manifold

orthochronous Lorentz group

orthogonal complement

orthogonal group

special

orthogonal vectors

orthonormal basis

standard order for

orthonormal frame

for pseudo-Riemannian metric

orthonormal vectors

osculating circle

outward-pointing normal

outward-pointing vector field

overdetermined system

P

parallel frame

parallel lines

parallel tensor field

parallel transport

along admissible curve

determines the connection

parallel vector field

parametrization

by arc length

graph

of a submanifold

parametrized curve

partition of an interval

partition of unity

path

path class

path-homotopic

path homotopy

path-lifting property

Perelman, Grigori

Pfaffian

piecewise regular curve segment

piecewise smooth vector field

pinching theorems

plane curve

plane curve classification theorem

plane section

Poincaré ball

Poincaré disk

Poincaré half-space

point reflection

pointwise conformal metrics

pointwise $\delta$ -pinched

pointwise pullback

polar coordinates

polar Fermi coordinates

polar normal coordinates

polarization identity

polygon

curved

geodesic

polynomial growth

positive density

positively oriented basis

positively oriented curved polygon

positively oriented $$n $$

-form

Preissman's theorem

Preissman, Alexandre

principal curvature comparison theorem

principal curvatures

principal directions

product metric

warped

product of paths

product rule

for connections

for divergence operator

for Euclidean connection

product, direct

projection

central

hyperbolic stereographic

normal

of a vector bundle

stereographic

tangential

projective plane

projective space

complex

real

proper group action

proper map

proper variation

proper vector field along a curve

properly embedded

pseudo-Euclidean space

pseudo-Riemannian geodesics

pseudo-Riemannian metric

pseudo-Riemannian normal coordinates

pseudo-Riemannian submanifold

pseudohyperbolic space

pseudosphere

pullback

of a connection

of a density

of an exterior derivative

pointwise

pullback orientation

pushforward of a vector field

Q

quadratic form

quotient manifold theorem

quotient map

R

Radó, Tibor

radial distance function

for a point

for a submanifold

radial geodesics

are minimizing

radial vector field

for a point

for a submanifold

raising an index

rank

constant

of a smooth map

of a tensor

of a vector bundle

rank theorem

equivariant

global

Rauch comparison theorem

ray in a Riemannian manifold

real projective space

regular coordinate ball

regular curve

regular domain

regular level set

regular point of a smooth map

regular value of a smooth map

related vector fields

relativity

general

special

reparametrization

backward

forward

of an admissible curve

representation

of a Lie algebra

of a Lie group

rescaling lemma

restricted exponential map

restricted holonomy

Riccati comparison theorem

matrix

Riccati equation

Riccati, Jacopo

Ricci curvature

geometric interpretation of

Ricci decomposition of the curvature tensor

Ricci flow

Ricci identities

Ricci tensor

geometric interpretation of

Riemann curvature endomorphism

Riemann curvature tensor

Riemann, Bernhard

Riemannian connection

Riemannian covering

Riemannian density

Riemannian distance

isometry invariance of

on a disconnected manifold

Riemannian geodesics

Riemannian isometry

Riemannian manifold

with boundary

Riemannian metric

existence of

Riemannian normal coordinates

Riemannian submanifold

of a pseudo-Riemannian manifold

Riemannian submersion

Riemannian symmetric space

locally

Riemannian volume form

is parallel

right action

right-invariant metric

right-invariant vector field

right translation

rigid motion

rotation index

of a curved polygon

rotation index theorem

for a curved polygon

rough section

round metric

S

SSS theorem

scalar curvature

geometric interpretation of

scalar product

scalar product space

scalar second fundamental form

Schoen, Richard

Schouten tensor

Schur's lemma

secant angle function

second Bianchi identity

second covariant derivative

second fundamental form

geometric interpretation of

scalar

second structure equation

second variation formula

section

of a submersion

of a vector bundle

smooth

sectional curvature

constant

determines the curvature tensor

of Euclidean space

of hyperbolic spaces

of spheres

segment, curve

self-dual

semi-Riemannian metric

semicolon between indices

semidirect product

semigeodesic coordinates

shape operator

sharp ( $\sharp$ )

sheets of a covering

side of a curved polygon

side-side-side theorem

sign conventions for curvature tensor

sign of a permutation

signature of symmetric bilinear form

signed curvature

of curved polygon

simple closed curve

simply connected space

covering of

single elliptic geometry

singular Riemannian metric

slice coordinates

boundary

smooth atlas

smooth chart

smooth coordinate ball

smooth coordinate chart

smooth coordinate domain

smooth covering map

smooth curve

smooth group action

smooth manifold

with boundary

smooth manifold structure

smooth map

smooth structure

smooth subbundle

smoothly compatible

solvable Lie group

soul theorem

space form

Euclidean

hyperbolic

spherical

special linear group

special orthogonal group

special relativity

special unitary group

spectral theorem

speed of a curve

sphere

Berger

distance function

geodesic

is locally conformally flat

principal curvatures of

round

sectional curvature

volume of

sphere theorem

differentiable

spherical coordinates

spherical space form

splitting theorem

standard basis of $\mathbb R ^n$

standard coordinates on $\mathbb{R }^n$

standard order for orthonormal basis

standard orientation of $\mathbb R ^n$

standard smooth structure

star operator

star-shaped

Steenrod, Norman E.

stereographic projection

hyperbolic

is a conformal diffeomorphism

Stiefel manifold

Stokes orientation

Stokes's theorem

stress-energy tensor

strictly $\delta$ -pinched

structure equation

first

second

sub-Riemannian metric

subalgebra, Lie

subbundle

subgroup, Lie

subinterval of a partition

submanifold

embedded

immersed

pseudo-Riemannian

Riemannian

tangent space to

with boundary

submersion

Riemannian

subordinate to a cover

summation convention

support

of a function

of a section of a vector bundle

surface of revolution

geodesics

Sylvester's law of inertia

symmetric connection

symmetric group

symmetric product

symmetric space

irreducible

locally

symmetric tensor

symmetries

of Euclidean space

of hyperbolic spaces

of spheres

of the curvature tensor

symmetrization

symmetry lemma

symplectic form

Synge's theorem

Synge, John L.

T

tangent angle function

tangent bundle

tangent covector

tangent cut locus

tangent space

to a submanifold

tangent to a submanifold

tangent vector

tangential acceleration

tangential connection

tangential directional derivative

tangential projection

tangential vector field along a curve

tensor

alternating

contravariant

covariant

mixed

on a manifold

symmetric

tensor bundle

tensor characterization lemma

tensor field

along a curve

tensor product

Theorema Egregium

Thurston geometrization conjecture

topological manifold

with boundary

torsion

-forms

torsion element of a group

torsion-free group

torsion tensor

torus

flat metric on

-dimensional

total covariant derivative

commutes with musical isomorphisms

components of

total curvature theorem

total scalar curvature functional

total space of a vector bundle

totally geodesic

trace

of a tensor

of an endomorphism

with respect to $$g $$

traceless Ricci tensor

transition function

transition map

transitive group action

on fibers

translation, left and right

transport, parallel

transverse curve

transverse Jacobi field

triangle

Euclidean

ideal

triangulation

trivial free homotopy class

tubular neighborhood

two-point boundary problem

U

Umlaufsatz

uniform tubular neighborhood

uniformization theorem

for compact surfaces

uniformly $\delta$ -normal

uniformly normal

unique lifting property

uniqueness of constant-curvature metrics

unit-speed curve

unit tangent bundle

unit tangent vector field

unitary group

special

universal anti-de Sitter space

universal covering manifold

upper half-plane

upper half-space

upper indices on coordinates

V

vacuum Einstein field equation

variation

first

normal

of a curve

proper

second

through geodesics

variation field

variational equation

variations, calculus of

vector bundle

section of

subbundle

zero section

vector field

along a curve

along a family of curves

canonical form for

commuting

coordinate

Lie algebra of

nonvanishing

normal, along a curve

piecewise smooth

proper

tangential, along a curve

vector, tangent

velocity

vertex

cusp

flat

of a curve

ordinary

vertical action

vertical index position

vertical tangent space

vertical vector field

volume

of a ball

of a Riemannian manifold

of a sphere

volume comparison theorem

Bishop–Gromov

Günther

volume form

von Mangoldt, Hans Carl Friedrich

W

Warner, Frank

warped product

wedge product

Alt convention

determinant convention

Weingarten equation

for a hypersurface

Weingarten map

Weyl–Schouten theorem

Weyl tensor

conformal invariance of

Wilking, Burkhard

Wolf, Joseph

Y

Yamabe equation

Yamabe problem

Z

zero section