Differential geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Dekker
1987
|
| Schriftenreihe: | Pure and applied mathematics
112 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 788 S. |
| ISBN: | 082477700X |
Internformat
MARC
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| 100 | 1 | |a Okubo, Tanjiro |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Differential geometry |c Tanjiro Okubo |
| 264 | 1 | |a New York [u.a.] |b Dekker |c 1987 | |
| 300 | |a XVIII, 788 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Pure and applied mathematics |v 112 | |
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
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| 830 | 0 | |a Pure and applied mathematics |v 112 |w (DE-604)BV000001885 |9 112 | |
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Datensatz im Suchindex
| _version_ | 1804115172355211264 |
|---|---|
| adam_text | Contents
PREFACE iii
Chapter 1
DIFFERENTIABLE MANIFOLDS 1
1. Differentiate Manifolds 1
1.1 Topological spaces and introductory material 1
1.2 Differentiate manifolds 6
1.3 Functions on manifolds 14
2. Vector Fields 19
2.1 Tangent vectors 19
2.2 Vector fields 23
2.3 Differentials of mappings 28
2.k Submanifolds 31
3. Tensor Fields 34
3.1 Tensor algebra 34
3.2 Tensor fields 40
3 3 Lie derivation and exterior differentiation 44
3.4 Distributions 49
h. Lie Groups and Lie Algebras (Part i) 51
^.1 Lie groups and Lie algebras 51
h.2 GL(n;g) and GL(n;C) 58
^ .3 Lie transformation groups 64
Chapter 2
THEORY OF CONNECTIONS 69
1. Fibre Bundles 69
1.1 Principal fibre bundles and associated fibre bundles 69
1.2 Reduction of structure groups 75
1.3 Covering manifolds 78
xiii
xiv Contents
2. Connections in Principal Fibre Bundles 80
2.1 Connection in P(M,G) 80
2.2 Parallel displacement and holonomy groups 88
2.3 Curvature form and structure equations 92
2.k Homomorphisms of connections 95
2.5 Holonomy theorem 97
2.6 Local holonomy groups 99
2.7 Infinitesimal holonomy groups 104
2.8 Flat connections 108
2.9 Connections in associated bundles 109
3. Linear Connections 110
3.1 Canonical 1 forms and connection forms 110
3.2 Covariant differentiation 113
3.3 Expression of canonical 1 forms and
connection forms in local coordinates 118
J.h Expression of covariant derivatives
in local coordinates 123
3.5 Bianchi identities 129
3.6 Linear holonomy groups 132
3.7 Affine connections 134
3.8 Development 137
3.9 Geodesies 141
3 10 Exponential mappings 144
3.11 Normal coordinates 147
3.12 Protective connections 150
3.13 Connections in vector bundles 158
Chapter 3
RIEMA.NNIAN MANIFOLDS l6l
1. Riemann Metrics and Riemann Connections l6l
1.1 Bundles of orthonormal frames
on Riemannian manifolds l6l
1.2 Riemann connections 168
2. Metric Properties of Riemannian Manifolds 177
2.1 Expression of Riemann connections
on orthonormal frames 177
2.2 Structure equations in polar systems 179
2.3 Completeness 190
3 Sectional Curvature and Spaces of Constant Curvature 195
3.1 Sectional curvature of a Riemannian manifold 195
3.2 Isometry and sectional curvature 203
3 3 Schur s theorem 214
3A Model spaces of spaces of constant curvature 219
3 5 Conformal transformations 231
k. Holonomy Groups of Riemannian Manifolds 236
k.1 Holonomy groups of pseudo Eiemannian manifolds 236
Contents xv
h.2 Decomposition and holonomy groups 238
It .3 Affine holonomy groups 245
5. Curvature Tensor Preserving Transformations 249
5 1 Curvature transformations and
invariant distributions 249
5 2 Curvature tensor preserving transformations 253
6. Leichtweiss Eiemann Metric of G(n,p) 256
6.1 Manifold structure of G(n,p) 256
6.2 Eiemannian structure of G(n,p) 259
Chapter It
THEOEY OF SUBMANIFOLDS 265
1. Bundles of Frames over Riemannian Submanifolds 265
1.1 Bundles of orthonormal frames over
Riemannian submanifolds and normal bundles 265
1.2 Expression of induced connections
in terms of orthonormal frames 270
2. Covariant Derivatives in Riemannian Submanifolds 272
2.1 Connecting quantities 272
2.2 Equations of Gauss and Weingarten 276
2.3 Gauss, Codazzi, and Ricci equations 282
2.it Absolute, relative, and normal curvatures 287
3. Isometric Immersions in Euclidean Spaces 292
3.1 Riemannian submanifolds of Euclidean spaces 292
3.2 Isometric embedding 299
3.3 Minimal immersions 310
k. The Gauss Map 319
It .l Gauss map Y G(n,p) in Rn P 319
h.2 Induced metrics on Gauss images 322
k.J Tension fields and harmonic Gauss maps 327
h.k Minimal Gauss map 329
5. Affine Submanifolds 337
5.1 Induced connections 337
5.2 Affine normals of affine hypersurfaces in A 440
Chapter 5
COMPLEX MANIFOLDS 347
1. Algebraic Preliminaries 347
1.1 Complexification and a complex structure
of a real vector space 347
xvi Contents
1.2 Decomposition of V° of V (= gj11)
Hermitian inner product 353
1.3 Functions on Cn 359
2. Complex Manifolds and Almost Complex Manifolds 36l
2.1 Complex manifolds 3 1
2.2 Almost complex manifolds 370
2.3 Integrability of a Cm almost complex structure 377
2.4 Almost complex affine connections 391
3. Metric Almost Complex Connections 395
3.1 Metric connections on almost Hermitian
and Hermitian manifolds 395
3.2 Expression in unitary frames of Riemann
connections on Kaehlerian manifolds 400
3.3 Expression of Riemann connections on a
Kaehlerian manifold in complex local coordinates 402
3 A Kaehlerian manifolds of constant
holomorphic sectional curvature 4l4
3.5 Kaehlerian submanlfolds 421
Chapter 6
HOMOGENEOUS AND SYMMETRIC SPACES 429
1. Lie Groups and Lie Algebras (Part II) 429
1.1 Complexification and real forms.
Nilpotent and solvable Lie algebras 429
1.2 Preliminary facts on representations 436
1.3 Representations of solvable and
nilpotent Lie algebras 441
1.4 Structure of semisimple Lie algebras 447
1.5 Weyl bases and compact real forms 458
1.6 Dynkin and extended Dynkin diagrams 464
2. Invariant Connections in Reductive Homogeneous Spaces 468
2.1 Reductive homogeneous spaces 468
2.2 Invariant affine connections 471
2.3 Canonical affine connections 474
2.4 Affine connections invariant under parallelism 481
3. Symmetric Spaces 486
3 1 Affine symmetric spaces 486
3 2 Symmetric spaces 490
3.3 Irreducible symmetric spaces 494
3 4 Riemannian symmetric spaces 495
3 5 Structure of the orthogonal involutive Lie algebra 497
3 d Sectional curvature of Riemannian symmetric
spaces. Minimal immersions 502
3.7 Totally geodesic submanifolds 507
Contents xvii
3.8 Hermitian symmetric spaces 510
3 9 Outline of classification of symmetric spaces 516
Chapter 7
G STRUCTURES AND TRANSFORMATION GROUPS 519
1. G Structures 519
1.1 Definition of G structures 519
1.2 G structures defined by tensors 521
1.3 G connections 525
l.k Prolongation and type number of linear Lie algebras 526
1.5 Jets and frames of higher order 530
1.6 Mdbius group K(n) 533
1.7 Cartan connections in P(Hn,K(n)) 539
1.8 Conformal structure and
normal conformal connections 542
2. Groups of Automorphisms 545
2.1 Automorphisms of G structures 545
2.2 Groups of automorphisms 546
3. Groups of Affine Transformations 551
3.1 Infinitesimal affine transformations 551
3.2 Integrability condition of £ V = 0 556
3 3 Classification of affinely connected manifolds
with torsion free connections by the order of
groups of affine transformations 562
k. Groups of Isometries on Riemannian Manifolds 572
i+.X Infinitesimal isometries 572
k.2 Structure of Riemannian manifolds admitting
groups of isometries of order r = (l/2)n(n l) + 1 576
Chapter 8
CALCULUS OF VARIATIONS FOR LENGTHS OF GEODESICS 589
1. Synge s Formula 589
1.1 Jacobi equations in affinely connected manifolds 589
1.2 Synge s formulas 594
1.3 Focal and conjugate points 599
l.k The Gauss lemma 608
2. Comparison Theorems 6ll
2.1 The Morse and Rauch comparison theorem 6ll
2.2 The Morse and Schoenberg comparison theorem 615
3. Cut Locus and the Index Theorem 620
3.1 Cut locus 620
3.2 Closed geodesies in lens spaces 62*4
3.3 Index theorem 626
xviii Contents
Chapter 9
THE DE EHAM THEOREM, CHARACTERISTIC CLASSES,
AND HARMONIC FORMS 637
1. de Rham Cohomology Theory 637
1.1 Integration over chains 637
1.2 de Rham cohomology 639
1.3 Sheaves and presheaves 645
l.k Fine and torsion free sheaves. Resolutions 65O
1.5 Cochain complexes 652
1.6 Abstract sheaf cohomology 654
1.7 Classical sheaf cohomologies 660
1.8 Proof of the de Rham theorem 672
2. Characteristic Classes 675
2.1 The Weil mapping 675
2.2 The Weil theorem 680
2.3 Invariant polynomials.
Special characteristic classes 684
3. Harmonic Functions and Forms 695
3.1 The Laplace Beltrami operator 695
3.2 Af = n f on compact and orientable
Riemannian manifolds 704
3.3 Decomposition theorem and the Hodge fundamental
theorem for compact and orientable Riemannian
manifolds 705
3.^ Curvature and Betti numbers 714
3.5 The operators L and A 721
3.6 Harmonic forms on Kaehlerian manifolds 727
3.7 Effective forms on Kaehlerian manifolds 732
NOTATION 741
BIBLIOGRAPHY 747
INDEX 78I
|
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| id | DE-604.BV000721813 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:18:23Z |
| institution | BVB |
| isbn | 082477700X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000451312 |
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| physical | XVIII, 788 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Dekker |
| record_format | marc |
| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Okubo, Tanjiro Verfasser aut Differential geometry Tanjiro Okubo New York [u.a.] Dekker 1987 XVIII, 788 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 112 Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Pure and applied mathematics 112 (DE-604)BV000001885 112 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000451312&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Okubo, Tanjiro Differential geometry Pure and applied mathematics Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012248-7 |
| title | Differential geometry |
| title_auth | Differential geometry |
| title_exact_search | Differential geometry |
| title_full | Differential geometry Tanjiro Okubo |
| title_fullStr | Differential geometry Tanjiro Okubo |
| title_full_unstemmed | Differential geometry Tanjiro Okubo |
| title_short | Differential geometry |
| title_sort | differential geometry |
| topic | Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000451312&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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