Abelian differential, 125
of first kind, 122
of first, second, and third kinds, 125
Abelian elementary differential,
of second kind, 120
of third kind, 122
Abelian function, 151
Abelian group, 67
Abel’s theorem, first version, 141, 142
second version, 149
Abstract definition of a surface, 30
Adaptation of a lattice, 89
Additive function, 131
Admissible coordinate system, 35
Admissible function, 97
Algebraic form, 155
Algebraic function field, 153
Analysis situs, 23
Analytic chain of function elements, 11
Analytic continuation; in particular,
along a curve, 2
on a Riemann surface, 43
Analytic curve, 39
Analytic differential, 61
Analytic form, 11
Analytic function, 4
Analytic neighborhood, 9
Angle, 40
Associated neighborhood, 36
Automorphic function, 169
Axiom of countability, 23
Base surface, 50
Basis of closed paths, 69
Basis of a field, 155
Basis of a linear family, 68
Bicontinuous map, 19
Birational transformation, 156
Block, 73
Bolyai-Lobatschefskian geometry, 170
Boundary, 18
Bounded, 21
Branch number, 150
Branch order, 8
Branch point, 150
Branched function element, 8
Brill-Noether reciprocity theorem, 139
Canonical basis, 90
Canonical pairs of cuts, 91
Cauchy integral theorem, 62
Center of a function element, 1
Class of Riemann surfaces, 171
Class of skew-symmetric forms, 90
Class of uniformizing variables, 166
Closed differential, 64
Closed set, 18
Closed surface, 24
Compact density, 74
Compact differential, 70
relative to the whole surface, 79
Compact set, 20
Complement, 18
Completely planar surface, 70
Component of an open set, 18
Conformally equivalent, 40
Conjugate differential form, 64, 93
Connected set, 18
Connectivity, degree of, 68, 70
continuous, 50
Continuous continuation, 50
Continuous curve, 18
Continuous density, 72
Continuous differential form (= continuous vector field), 63
Continuous function, 18
Continuous map, 19
Continuously differentiable curve, 40
Continuously differentiable function, 36
Convergence disc, 1
Convex set, 45
Cover transformation, 57
Covering surface, 50
Critical point, 3
function of, 66
Cut-circle uniformizing variable, 170
Cycle of vertices, 173
Degree, of a field, 155
Density, 72
Determining function of an algebraic field, 154
Dieudonné factors, 73
Differentiable map, 35
Dirichlet integral, 94
Dirichlet principle, 94
Discontinuous group, 169
Distance, non-Euclidean, 171
Divisor, 135
Doubly periodic function, 41, 143
Elementary differential, 120, 122
Elementary subdivision of a square, 24
Elliptic transformation, 179
Equivalent, 5, 23, 28, 40, 90, 125
Essential singularity, 38
Exact differential, 64
Exact function of curves, 66
Field, 153
Field equation, 154
Fixed point, 169
Function, 18
determining an algebraic field, 154
in the special sense of “meromorphic function on a Riemann
surface”, 43
integral, 66
Function element, branched and unbranched, 8
in general sense, 5
in narrow sense, 1
regular, 7
Fundamental group, 58
Fundamental region, 171
Galois group, 58
Gap theorem, 140
Generalized Riemann-Roch theorem, 147
Genus, 89
Green’s formula, 78
Group of cover transformations, 57, 58
Group of local coordinate transformations, 35
Harmonic differential, 93
Harmonic function, 39
elementary of the second kind, 120
Heine-Borel theorem, 21
Hull, 18
Hyperbolic transformation, 179
Hyperelliptic, 157
Image, of a curve, 19
of a set, 19
Imbedding, 50
Immediate analytic continuation, 2
Improper integral, 77
Increment function, 66
Infinitesimal transformation, 179
Integral, of a closed differential along a curve, 65
of a density, 73-76
Integral basis, 89
Integral characters, of a divisior, 140 141
of a point, 140
Integral function, 66
Interior point, 18
Intersection, 18
Inversion problem, 144, 151-152
Irreducible polynomial, 154
Isolated point, 18
Jordan content, 76
Kernel of a set, 18
Left and right, 79
Lid, 107
Lies over, 50
Linear dependence, 68
Linear differential form, 63
Linear function of curves, 66
Lobatschefskian geometry, 170
Local coordinate system, 34
Local map, 34
Local (uniformizing) parameter, 32, 36
Locally not distinct, 34
Lock-ring, 107
Manifold; in particular, two-dimensional, 16–17
Map, 19
Mediate analytic continuation, 2
Meromorphic function, 43
Modified cohomology, 83
Module of a Riemann surface, 41
Möbius strip, 27
Monodromy theorem, 59
Motion of the non-Euclidean plane, 170
Multiple of a divisor, 135
Multiplicative differential, 146
Multiplicative function, 132
elementary, 133
Multiplier, of a multiplicative function, 132
of a transformation, 179
Negative map, differentiable, 35
topological, 47
Neighborhood, 17
analytic, 9
Neighborhood true, 19
Non-Euclidean distance, 171
Non-Euclidean motion, 170
Non-Euclidean plane, 171
Normal covering surface, 57
Normal form of a Riemann surface, 169
Normal polygon, 173
Normal representation of a function element, 7-8
ω-function, 82
One-sheeted covering surface, 52
One-(to)-one map, 19
Order, of a branch-point, 8, 149-150
of a differential at a point, 61
(total) of a divisor, 135
of a function at a point, 37–38
of a group, 182
of a point relative to a curve, 44
Orientable surface, 54
Oriented smooth or topological surface, 36, 47
Orthogonal projection, 110
Parabolic transformation, 179
Parameter, local (uniformizing), 32, 36
Perfect covering surface, 52
Periods, 119
Place, 50
Planar surface, 70
completely, 70
Plane, non-Euclidean, 171
projective, 27
Poincaré 
-series, 177
Point lattice, 30
Poisson integral, 97
Pole, of a differential, 61
of a function, 38
Positive map, differentiable, 35
topological, 47
Positive sense of rotation, 44, 47
Potential function, 39
Primitive closed path, 91
Principal character, 146
Principal part,of an additive function, 132
of a differential, 125
Probability function, 74
Projective plane, 27
Prym differential, 146
Punched surface, 107
Quadratic composition, rule of, 96
Regular function element, 7
Representation of a function element, 5
Residue theorem, 79
Reversibly neighborhood true map, 20
Riemann surface, 36
Riemann-Roch theorem, 137
generalized, 147
Right and left, 79
Ring, 156
Rotation of the non-Euclidean plane, 171
Rule of quadratic composition, 96
Scarp function, 76
Scattered set, 22
Schwarz’ lemma, 168
Sense of rotation, on smooth surfaces, 44
on topological surfaces, 47
Sheets of a covering surface, 52
Simply connected, 52
Singularity of a function, 38
Skew product, 72
Smooth surface, 36
Smoothing function, 71
Standard subdivision of a curve, 22
Stokes-Green formula, 78
Stream, 68
Surface, 24
System of equivalent points, 28, 169
t-neighborhood, 9
-series, 177
Topological map, 20
Topologically equivalent, 23
Topology, 23
Torus, 29
Trace point, 50
Transcendental normed Abelian differential, (or integral) of the second kind, 131
Triangulation, 25
Two dimensional, 17
Two-sided surface, 54
Type of a surface, 35
Unbranched covering surface, 50
Unbranched function element, 8
Uniform convergence except at the boundary, 99
Uniformization principle, 166
Uniformizing variable, 159
cut-circle, 170
Unimodular transformation, 89
Union, 18
Unit disc, 17
Universal covering surface, 58
Unlimited covering surface, 51
Validity of a representation, 8
Vector (and vector field), 63–64
Vertex, 173
Weak degree of connectivity, 70
Weak homology, 70
Weierstrass gap theorem, 140
Zero, of a differential, 60
of a function, 37