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| Helgason S. — Differential Geometry, Lie Groups and Symmetric Spaces |
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| Предметный указатель |
Abelian Lie algebra 100
Adapted vector field 36
Adjoint group 126
Adjoint operator 143
Adjoint representation 100 127
Adjoint space 327
Affine connection 26
Affine transformation 28
Affine Weyl group 314
Affine, locally symmetric space 198
Almost complex, manifold 352
Almost complex, mapping 355
Almost complex, structure 352
Alpha-series 168
Alternate 13
Alternation 18
Analytic affine connection 36
Analytic diffeomorphism 22
Analytic function 5
Analytic homomorphism 99
Analytic isomorphism 99
Analytic manifold 4
Analytic mapping 2
Analytic pseudo-Riemannian manifold 48
Analytic Riemannian manifold 205
Analytic structure 4
Analytic subgroup 112
Analytic vector field 11
Antipodal set 328
Automorphism inner 127
Automorphism of Dynkin diagram 478 505
Automorphism of root system 421 456
Automorphism of type  510
Automorphism outer 421 478
Ball 51
Basis compatible 264
Basis of a root system 458
Basis of vector fields 10 44
Bergman metric 369
Bi-invariant 135
Bianchi identity 69
Bounded domain 364
Bounded symmetric domain 382
Bruhat decomposition 403
Canonical coordinates 104
Canonical coordinates of the second kind 146
Canonical generators 482
Canonical polynomial 264
Canonical, almost complex structure 355
Cartan decomposition 183
Cartan involution 184
Cartan matrix 459 496
Cartan subalgebra 162
Category theorem 120
Cauchy sequence 55
Cayley transform 95 531
cell 314 321
Center 100 311
centralizer 99 284
Centroid 500
Chain 466
Chain condition 158
Character of a real form 517
Characteristic equation 156
Characteristic polynomial 156
Christoffel symbols 95
Clifford — Wolf isometry 279
Closed form 136
Compact Lie algebra 130
Compact open topology 202
Compact type 230 231
Compactly imbedded subalgebra 130
Complete affine connection 149
Complete Riemannian manifold 55
Completely reducible 192
Completeness at a point 58
Complex 1-form 356
Complex Lie group 392
Complex manifold 4
Complex structure on a Lie algebra 179
Complex structure on a topological space 353
Complex structure on a vector space 178
Complex tangent vectors 357
Complex tensor field 356
Complex vector field 356
Complex vector field, type of 357
Complexification of a Lie algebra 180
Complexification of a vector space 179
Conjugacy 183
Conjugate point 59 294 303
Conjugation 180
Connected 466
Continuation of an isometry 62
Contraction 17
Contravariant 13
Convex neighborhood 34
Convex normal ball 54
Coordinate neighborhood 4
Covariant derivative 42
Covariant differentiation 26
Covariant type 14
Covering group 98
Covering homomorphism 491
covering index 479
Covering Lie algebra 491
Covering manifold 59
Covering of a space 8
Covering space 59
Covering transformation 310
Critical point 279
Curvature 64
Curvature tensor field 44
Curve 28
Curve segment 28
CYCLE 466
de Rham’s Theorem 1
Decomposable 496
Degree of a polynomial mapping 267
Derivation 8
Derivation inner 127
Derived algebra 158
Diagonal matrix 155
Diagram of a compact group 299
Diagram of a symmetric space 295
Diffeomorphism 2 22
Differentiable function 5 30
Differentiable manifold 4
Differentiable mapping 2 22
Differentiable structure 3
Differential 22
Differential form 11
Differential form, bi-invariant 135
Differential form, left invariant 135
Differential form, right invariant 135
Direct product 13
Direct sum 13
Displacement function 278
Dual 11 235
Dynkin diagram 462
Dynkin diagram, extended 505
Effective transformation group 120
Eigenspace 155
Eigenvalue 155
Eigenvector 156
Elliptic 430
Engel’s theorem 160
Exponential mapping 33 104
Expression of a form 20
Expression of a mapping 22
Extendable isometry 62
| Exterior differential forms 17
Exterior differentiation 19
Exterior product 19
Flat Riemannian manifold 245
Flat totally geodesic submanifold 245
Free Lie algebra 482
Fundamental group 307 310
Geodesic 29
Geodesic closed, curvature 93
Geodesic manifold 79
Geodesic polar coordinates 95
Geodesic simply closed 327
Geodesic symmetry 198
Gradation 490
Gradation of type  509
Graph of a mapping 91
Graph of a root system 461
Grassmann algebra 19
Harmonic form 144 227
Height 460
Hermitian matrix 444
Hermitian structure 358
Hermitian symmetric space 372
Holomorphic 4
Holomorphic function 4
Holomorphic mapping 353
Holonomy group 197 193
Homogeneous complex manifold 371
Homogeneous space 120
Homomorphism analytic 99
Homomorphism of a Lie algebra 99
Hyperbolic element 430
Hyperbolic plane 93
Hyperbolic space Hermitian 452
Hyperbolic space quaternian 455
Hyperbolic space real 453
Ideal 99
Ideal, graded 490
Identity component 98
Immediate continuation 62
Imprimitive 151
Indecomposable 496
Indivisible 456
Integrable 353
Integral curve 40
Interior product 40
Invariant, affine connection 28
Invariant, subspace 156
Invariant, tensor field 25
Involution 205 229
Irreducible orthogonal symmetric Lie algebra 377 379
Irreducible pair 377
Irreducible Riemannian globally symmetric space 378
Irreducible root system 458
Irreducible subspace 156
Isometry 60
Isomorphic 109 230 438
Isomorphism 99
Isotropic manifold 535
Isotropy group 121 125
Isotropy subgroup 121
Iwasawa decomposition 260 263 270 273 403 435
Jacobi equation 71
Jacobi identity 9
Jordan decomposition 431
Jordan decomposition, additive 431
Jordan decomposition, complete multiplicative 431
Jordan decomposition, multiplicative 431
Kaehlerian structure 358
Kernel function 367
Killing form 131
Killing form of a root system 523
Laplace — Beltrami operator 92
Left invariant, affine connection 102
Left invariant, differential form 135
Left invariant, vector field 99
length function 315
Lexicographic ordering 173
Lie algebra of a Lie group 99
Lie algebra over a field 99
Lie algebra product 100
Lie derivative 9 89
Lie group 98
Lie subgroup 112
Lie theorem 159
Lie transformation group 121
Lie triple system 224
Lift 59 320
Linear isotropy group 125
Link 466
Link double 466
Link simple 466
Link triple 466
Local chart 4
Local coordinates 4
Local cross section 123
Local isometry 60
Locally finite 8
Locally isomorphic 109
Locally nilpotent 488
Locally, Euclidean 98
Lower triangular 155
Manifold 4
Matrix exponential function 111
Maurer — Cartan equations 137
Maximal compact subgroups 256
Midpoint locus 341
Minimizing ball 54
Mixed tensor algebra 16
Monothetic 287
Multilinear 13
Multiplicity 264 428
Natural topology 120
Negative curvature 72
Nilpotent endomorphism 156
Nilpotent endomorphism, Lie algebra 160
Noncompact type 230 231 252
Normal coordinates 33
Normal coordinates neighborhood 33
Normal coordinates, topological space 8
Normal coordinatesreal form 426
normalizer 99 282 284
One-parameter subgroup 104
Open chart 3
Open chart, submanifold 7
Ordered set 172
Ordered set, vector space 172
Orderings compatible 260
Orientable 361
Oriented 361
Orthogonal symmetric Lie algebra 213
Orthogonal symmetric Lie algebra a pair associated with 213
Orthogonal symmetric Lie algebra, compact type 230
Orthogonal symmetric Lie algebra, decomposition of 231 378
Orthogonal symmetric Lie algebra, duality for 235
Orthogonal symmetric Lie algebra, effective 230
Orthogonal symmetric Lie algebra, isomorphism of 230
Orthogonal symmetric Lie algebra, noncompact type 230
pair 213
Paracompact 8 82
parallel 29
Partition of unity 8
Phi-related 24
Polar coordinate decomposition 402
Polynomial function 267
Polynomial function mapping, degree of 267
Polynomial function, mapping 267
Positive definite 526
primitive 151
Product manifold 7
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