Riemannian geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1993
|
| Ausgabe: | 2. ed., corr. 2. print. |
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 284 S. graph. Darst. |
| ISBN: | 3540524010 0387524010 |
Internformat
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| 100 | 1 | |a Gallot, Sylvestre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Riemannian geometry |c Sylvestre Gallot ; Dominique Hulin ; Jacques Lafontaine |
| 250 | |a 2. ed., corr. 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1993 | |
| 300 | |a XIII, 284 S. |b graph. Darst. | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-018467198 | ||
Datensatz im Suchindex
| _version_ | 1804140485718048768 |
|---|---|
| adam_text | Titel: Riemannian geometry
Autor: Gallot, Sylvestre
Jahr: 1993
CONTENTS
Chapter I. Differential Manifolds
A. From Submanifolds to Abstract Manifolds
Submanifolds of Rn+t ......................................................................................................2
Abstract manifolds ............................................................................................................6
Smooth maps ......................................................................................................................11
B. Tangent Bundle
Tangent space to a submanifold of Rn+* ..................................................................13
The manifold of tangent vectors ................................................................................14
Vector bundles ..................................................................................................................16
Differential map ................................................................................................................17
C. Vector Fields
Definitions ..........................................................................................................................18
Another definition for the tangent space ................................................................19
Integral curves and flow of a vector field ................................................................23
Image of a vector field under a diffeomorphism ....................................................24
D. Baby Lie Groups
Definitions ..........................................................................................................................27
Adjoint representation ....................................................................................................29
E. Covering Maps and Fibrations
Covering maps and quotient by a discrete group ..................................................29
Submersions and fibrations ..........................................................................................31
Homogeneous spaces ........................................................................................................33
F. Tensors
Tensor product (digest) ..................................................................................................36
Tensor bundles ..................................................................................................................36
Operations on tensors ....................................................................................................37
Lie derivatives ....................................................................................................................39
Local operators, differential operators ......................................................................40
A characterization for tensors ......................................................................................40
X
Contents
G. Exterior Forms
Definitions .............................................................
Exterior derivative ......................................................
Volume forms ..........................................................
Integration on an oriented manifold .....................................
Haar measure on a Lie group ...........................................
H. Appendix: Partitions of Unity .................................
Chapter II. Riemannian Metrics
A. Existence Theorems and First Examples
Definitions .............................................................
First examples ..........................................................
Examples: Riemannian submanifolds, product Riemannian manifolds ----
Riemannian covering maps, flat tori .....................................
Riemannian submersions, complex projective space ......................
Homogeneous Riemannian spaces ........................................
B. Covariant Derivative
Connections .............................................................
Canonical connection of a Riemannian submanifold ......................
Extension of the covariant derivative to tensors ..........................
Covariant derivative along a curve .......................................
Parallel transport .......................................................
Examples ...............................................................
C. Geodesies
Definitions ..............................................................
Local existence and uniqueness for geodesies, exponential map ...........
Riemannian manifolds as metric spaces ..................................
Complete Riemannian manifolds, Hopf-Rinow theorem ...................
Geodesies and submersions, geodesies of PnG ...........................
Cut locus ..............................................................
Chapter III. Curvature
A. The Curvature Tensor
Second covariant derivative .............................................
Algebraic properties of the curvature tensor ............................
Computation of curvature: some examples
Ricci curvature, scalar curvature ........................................
B. First and Second Variation of Arc-Length and Energy
Technical preliminaries:
vector fields along parameterized submanifolds .........................
42
43
46
47
48
48
52
54
58
59
63
65
69
72
73
75
77
78
80
83
87
94
97
100
107
108
109
111
112
Contents XI
First variation formula ..................................................................................................114
Second variation formula ..............................................................................................116
C. Jacobi Vector Fields
Basic topics about second derivatives ......................................................................118
Index form ..........................................................................................................................119
Jacobi fields and exponential map ............................................................................121
Applications: Sn, Hn, PnR, 2-dimensional Riemannian manifolds ............122
D. Riemannian Submersions and Curvature
Riemannian submersions and connections ..............................................................124
Jacobi fields of P C ......................................................................................................125
O Neill s formula ..............................................................................................................127
Curvature and length of small circles.
Application to Riemannian submersions ................................................................128
E. The Behavior of Length and Energy
in the Neighborhood of a Geodesic
The Gauss lemma ............................................................................................................130
Conjugate points ..............................................................................................................131
Some properties of the cut-locus ................................................................................134
F. Manifolds with Constant Sectional Curvature
Spheres, Euclidean and hyperbolic spaces ............................................................135
G. Topology and Curvature
The Myers and Hadamard-Cartan theorems ........................................................137
H. Curvature and Volume
Densities on a differentiable manifold ......................................................................139
Canonical measure of a Riemannian manifold ......................................................140
Examples: spheres, hyperbolic spaces, complex projective spaces ................142
Small balls and scalar curvature ................................................................................143
Volume estimates ............................................................................................................144
I. Curvature and Growth of the Fundamental Group
Growth of finite type groups ...........................................- 148
Growth of the fundamental group of compact manifolds
with negative curvature ................................................................................................149
J. Curvature and Topology:
An Account of Some Old and Recent Results
Introduction ......................................................................................................................151
Traditional point of view: pinched manifolds ........................................................151
Almost flat pinching ........................................................................................................153
Coarse point of view: compactness theorems of Cheeger and Gromov .... 153
XII
Contents
K. Curvature Tensors and Representations
of the Orthogonal Group
Decomposition of the space of curvature tensors ........................ 154
Conformally flat manifolds .............................................
The second Bianchi identity ............................................
L. Hyperbolic Geometry
159
Introduction ..........................................* *...............
Angles and distances in the hyperbolic plane ...........................
Polygons with many right angles .................................... j64
Compact surfaces ......................................................
Hyperbolic trigonometry ...............................................
Prescribing constant negative curvature ................................ 172
M. Conformal Geometry
Introduction ......................................................................................................................*74
The Moebius group ........................................................................................................1^
Conformal, elliptic and hyperbolic geometry ........................................................177
Chapter IV. Analysis on Manifolds
and the Ricci Curvature
A. Manifolds with Boundary
Definition ............................................................. 181
The Stokes theorem and integration by parts ........................... 182
B. Bishop s Inequality Revisited
Some commutations formulas ......................................................................................185
Laplacian of the distance function ............................................................................186
Another proof of Bishop s inequality ........................................................................187
The Heintze-Karcher inequality ..................................................................................188
C. Differential Forms and Cohomology
The de Rham complex ..................................................................................................190
Differential operators and their formal adjoints ..................................................190
The Hodge-de Rham theorem ....................................................................................193
A second visit to the Bochner method ....................................................................194
D. Basic Spectral Geometry
The Laplace operator and the wave equation ........................... 196
Statement of the basic results on the spectrum .......................... 198
E. Some Examples of Spectra
Introduction ............................................jgg
The spectrum of flat tori ........................................................................200
Spectrum of (Sn, can) .................................................................................9(U
Contents XIII
F. The Minimax Principle
The basic statements ......................................................................................................203
G. The Ricci Curvature and Eigenvalues Estimates
Introduction ......................................................................................................................207
Bishop s inequality and coarse estimates ................................................................207
Some consequences of Bishop s theorem ................................................................208
Lower bounds for the first eigenvalue ......................................................................210
H. Paul Levy s Isoperimetric Inequality
Introduction ......................................................................................................................212
The statement ......................................................... 212
The proof ............................................................................................................................213
Chapter V. Riemannian Submanifolds
A. Curvature of Submanifolds
Introduction ......................................................................................................................217
Second fundamental form ............................................................................................217
Curvature of hypersurfaces ..........................................................................................219
Application to explicit computations of curvatures ............................................221
B. Curvature and Convexity
The Hadamard theorem ..............................................................................................224
C. Minimal Surfaces
First results ........................................................................................................................227
Some Extra Problems ............................................................................................232
Solutions of Exercises
Chapter I ............................................................................................................................234
Chapter II ..........................................................................................................................244
Chapter III .......................................................................................261
Chapter IV ........................................................................................................................266
Chapter V .................................................................................................................268
Bibliography ................................................................................................................272
Index ................................................................................................................................279
|
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| id | DE-604.BV024492054 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:00:44Z |
| institution | BVB |
| isbn | 3540524010 0387524010 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018467198 |
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| physical | XIII, 284 S. graph. Darst. |
| publishDate | 1993 |
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| publisher | Springer |
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| series2 | Universitext |
| spelling | Gallot, Sylvestre Verfasser aut Riemannian geometry Sylvestre Gallot ; Dominique Hulin ; Jacques Lafontaine 2. ed., corr. 2. print. Berlin [u.a.] Springer 1993 XIII, 284 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s DE-604 Hulin, Dominique Sonstige oth Lafontaine, Jacques Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018467198&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gallot, Sylvestre Riemannian geometry Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4128462-8 |
| title | Riemannian geometry |
| title_auth | Riemannian geometry |
| title_exact_search | Riemannian geometry |
| title_full | Riemannian geometry Sylvestre Gallot ; Dominique Hulin ; Jacques Lafontaine |
| title_fullStr | Riemannian geometry Sylvestre Gallot ; Dominique Hulin ; Jacques Lafontaine |
| title_full_unstemmed | Riemannian geometry Sylvestre Gallot ; Dominique Hulin ; Jacques Lafontaine |
| title_short | Riemannian geometry |
| title_sort | riemannian geometry |
| topic | Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Riemannsche Geometrie |
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