Holdings: Riemannian geometry

Riemannian geometry:

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Main Author:	Gallot, Sylvestre (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 1993
Edition:	2. ed., corr. 2. print.
Series:	Universitext
Subjects:	Riemannsche Geometrie
Online Access:	Inhaltsverzeichnis
Physical Description:	XIII, 284 S. graph. Darst.
ISBN:	3540524010 0387524010

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MARC


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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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isbn	3540524010 0387524010
language	English
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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
url	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
work_keys_str_mv	AT gallotsylvestre riemanniangeometry AT hulindominique riemanniangeometry AT lafontainejacques riemanniangeometry

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