Index
313
Partition of unity, 267, 268
Path, 10
– admissible or of class C
1/2 , 10
– exterior of a path, 32
– index of a point, 32
– interior of a path, 32
– length of a path, 34
– lifting, 288
– support, 32
Primitive
– global existence, 44
– of a differential form, 166
– of a holomorphic function, 8
Principal value, 92
Rank of a map at a point, 148
Regulated
– function, 3
Residue, 42
– at infinity, 50
– conformal invariance, 51
– of a differential form, 52, 278
Riemann sphere, 54
Riemann surface, 275
– and elliptic functions, 276
– genus, 280
– holomorphic (differential), 277
– holomorphic (function), 276
– holomorphic chart, 275
– local uniformizer, 275
– meromorphic (differential), 277
– meromorphic (function), 276
– of an algebraic function, 283, 298–300
– order at a point, 277, 279
– residue at a point, 278
Section (of a covering space), 287
Space
– Cartesian, 135
– projective, 222
– Riemann, 163–165, 248
Star (domain), 9
Subimmersion, 155, 233
Submanifold, 237
– defined by a submersion, 239
– defined by an immersion, 240
– of a Cartesian space, 246
Submersion, 155, 233
– canonical form, 156
Tangent (linear map), 147
Tangent vector
– at a point of a manifold, 226
– manifold of tangent vectors, 233
– to a Cartesian space, 229, 231
– to a curve, 228
Tensor, 136, 159–161
– at a point of a manifold, 226
– covariant, contravariant, 136
– Einstein’s summation convention,
137
– of type (p, q), 136
– tensor notation, 135–139
– tensor product, 136
Transpose, 134
Uniform (branch), 283, 297, 298
Vector field, 250
– integral curves, 252
THEOREMS of Chap. VIII
– Theorem 1, 15
– Theorem 2, 18
– Theorem 3, 25
– Theorem 4, 31
– Theorem 5, 44
– Theorem 6, 46
– Theorem 7, 49
– Theorem 8, 56
– Theorem 9, 59
– Theorem 10, 93
– Theorem 11, 94
– Theorem12, 97
– Theorem 13, 103
– Theorem 14, 111
THEOREMS of Chap. IX
– Theorem 1, 156
– Theorem 2, 173
– Theorem 3, 177
THEOREMS of Chap. X
– Theorem 1, 278
– Theorem 2, 288
– Theorem 3, 292
– Theorem 4, 293
– Theorem 5, 296
– Theorem 6, 297
– Theorem 7, 303
– Theorem 8, 305
313
Partition of unity, 267, 268
Path, 10
– admissible or of class C
1/2 , 10
– exterior of a path, 32
– index of a point, 32
– interior of a path, 32
– length of a path, 34
– lifting, 288
– support, 32
Primitive
– global existence, 44
– of a differential form, 166
– of a holomorphic function, 8
Principal value, 92
Rank of a map at a point, 148
Regulated
– function, 3
Residue, 42
– at infinity, 50
– conformal invariance, 51
– of a differential form, 52, 278
Riemann sphere, 54
Riemann surface, 275
– and elliptic functions, 276
– genus, 280
– holomorphic (differential), 277
– holomorphic (function), 276
– holomorphic chart, 275
– local uniformizer, 275
– meromorphic (differential), 277
– meromorphic (function), 276
– of an algebraic function, 283, 298–300
– order at a point, 277, 279
– residue at a point, 278
Section (of a covering space), 287
Space
– Cartesian, 135
– projective, 222
– Riemann, 163–165, 248
Star (domain), 9
Subimmersion, 155, 233
Submanifold, 237
– defined by a submersion, 239
– defined by an immersion, 240
– of a Cartesian space, 246
Submersion, 155, 233
– canonical form, 156
Tangent (linear map), 147
Tangent vector
– at a point of a manifold, 226
– manifold of tangent vectors, 233
– to a Cartesian space, 229, 231
– to a curve, 228
Tensor, 136, 159–161
– at a point of a manifold, 226
– covariant, contravariant, 136
– Einstein’s summation convention,
137
– of type (p, q), 136
– tensor notation, 135–139
– tensor product, 136
Transpose, 134
Uniform (branch), 283, 297, 298
Vector field, 250
– integral curves, 252
THEOREMS of Chap. VIII
– Theorem 1, 15
– Theorem 2, 18
– Theorem 3, 25
– Theorem 4, 31
– Theorem 5, 44
– Theorem 6, 46
– Theorem 7, 49
– Theorem 8, 56
– Theorem 9, 59
– Theorem 10, 93
– Theorem 11, 94
– Theorem12, 97
– Theorem 13, 103
– Theorem 14, 111
THEOREMS of Chap. IX
– Theorem 1, 156
– Theorem 2, 173
– Theorem 3, 177
THEOREMS of Chap. X
– Theorem 1, 278
– Theorem 2, 288
– Theorem 3, 292
– Theorem 4, 293
– Theorem 5, 296
– Theorem 6, 297
– Theorem 7, 303
– Theorem 8, 305