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Geometry VI : Riemannian Geometry :  Riemannian Geometry - M. M. Postnikov

Geometry VI : Riemannian Geometry

Riemannian Geometry

Hardcover Published: April 2001
ISBN: 9783540411086
Number Of Pages: 504

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This book treats that part of Riemannian geometry related to more classical topics in a very original, clear and solid style. Before going to Riemannian geometry, the author presents a more general theory of manifolds with a linear connection. Having in mind different generalizations of Riemannian manifolds, it is clearly stressed which notions and theorems belong to Riemannian geometry and which of them are of a more general nature. Much attention is paid to transformation groups of smooth manifolds. Throughout the book, different aspects of symmetric spaces are treated. The author successfully combines the co-ordinate and invariant approaches to differential geometry, which give the reader tools for practical calculations as well as a theoretical understanding of the subject. The book contains a very useful large Appendix on foundations of differentiable manifolds and basic structures on them which makes it self-contained and practically independent from other sources.

Industry Reviews

From the reviews of the first edition:

"... The book is .. comprehensive and original enough to be of interest to any professional geometer, but I particularly recommend it to the advanced student, who will find a host of instructive examples, exercises and vistas that few comparable texts offer... "

H.Geiges, Nieuw Archief voor Wiskunde 2002, Vol. 5/3, Issue 4

"... I found the presentation insightful and stimulating. A useful paedagogical device of the text is to make much use of both the index and coordinate-free notations, encouraging flexibility (and pragmatism) in the reader. ... the book should be of use to a wide variaty of readers: the relative beginner, with perhaps an introductory course in differential geometry, will find his horizons greatly expanded in the material for which this prepares him; while the more experienced reader will surely find the more specialised sections informative."

Robert J. Low, Mathematical Reviews, Issue 2002g

"... Insgesamt liegt ein sehr empfehlenswertes Lehrbuch einerseits zur Riemannschen Geometrie und andererseits zur Theorie differenzierbarer Mannigfaltigkeiten vor, wegen der strukturierten Breite der Darstellung sehr gut geeignet sowohl zum Selbststudium fur Studierende mathematischer Disziplinen als auch fur Dozenten als Grundlage einschlagiger Lehrveranstaltungen."

P. Paukowitsch, Wien (IMN - Internationale Mathematische Nachrichten 190, S. 64-65, 2002)

"M.M. Postnikov has written a well-structured and readable book with a satisfying sense of completeness to it. The reviewer believes this book deserves a place next to the already existing literature on Riemannian geometry, principally as a basis for teaching a course on abstract Riemannian geometry (after an introduction to differentiable manifolds) but also as a reference work." (Eric Boeckx, Zentralblatt MATH, Vol. 993 (18), 2002)

Prefacep. V
Affine Connectionsp. 1
Connection on a Manifoldp. 1
Covariant Differentiation and Parallel Translation Along a Curvep. 3
Geodesicsp. 4
Exponential Mapping and Normal Neighborhoodsp. 7
Whitehead Theoremp. 9
Normal Convex Neighborhoodsp. 13
Existence of Leray Coveringsp. 13
Covariant Differentiation. Curvaturep. 14
Covariant Differentiationp. 14
The Case of Tensors of Type (r, 1)p. 16
Torsion Tensor and Symmetric Connectionsp. 18
Geometric Meaning of the Symmetry of a Connectionp. 20
Commutativity of Second Covariant Derivativesp. 21
Curvature Tensor of an Affine Connectionp. 22
Space with Absolute Parallelismp. 24
Bianci Identitiesp. 24
Trace of the Curvature Tensorp. 27
Ricci Tensorp. 27
Affine Mappings. Submanifoldsp. 29
Affine Mappingsp. 29
Affinitiesp. 32
Affine Coveringsp. 33
Restriction of a Connection to a Submanifoldp. 35
Induced Connection on a Normalized Submanifoldp. 37
Gauss Formula and the Second Fundamental Form of a Normalized Submanifoldp. 38
Totally Geodesic and Auto-Parallel Submanifoldsp. 40
Normal Connection and the Weingarten Formulap. 42
Van der Waerden-Bortolotti Connectionp. 42
Structural Equations. Local Symmetriesp. 44
Torsion and Curvature Formsp. 44
Cartan Structural Equations in Polar Coordinatesp. 47
Existence of Affine Local Mappingsp. 50
Locally Symmetric Affine Connection Spacesp. 51
Local Geodesic Symmetriesp. 53
Semisymmetric Spacesp. 54
Symmetric Spacesp. 55
Globally Symmetric Spacesp. 55
Germs of Smooth Mappingsp. 55
Extensions of Affine Mappingsp. 56
Uniqueness Theoremp. 58
Reduction of Locally Symmetric Spaces to Globally Symmetric Spacesp. 59
Properties of Symmetries in Globally Symmetric Spacesp. 60
Symmetric Spacesp. 61
Examples of Symmetric Spacesp. 62
Coincidence of Classes of Symmetric and Globally Symmetric Spacesp. 63
Connections on Lie Groupsp. 67
Invariant Construction of the Canonical Connectionp. 67
Morphisms of Symmetric Spaces as Affine Mappingsp. 69
Left-Invariant Connections on a Lie Groupp. 70
Cartan Connectionsp. 71
Left Cartan Connectionp. 73
Right-Invariant Vector Fieldsp. 74
Right Cartan Connectionp. 76
Lie Functorp. 77
Categoriesp. 77
Functorsp. 78
Lie Functorp. 79
Kernel and Image of a Lie Group Homomorphismp. 80
Campbell-Hausdorff Theoremp. 82
Dynkin Polynomialsp. 83
Local Lie Groupsp. 84
Bijectivity of the Lie Functorp. 85
Affine Fields and Related Topicsp. 87
Affine Fieldsp. 87
Dimension of the Lie Algebra of Affine Fieldsp. 89
Completeness of Affine Fieldsp. 91
Mappings of Left and Right Translation on a Symmetric Spacep. 94
Derivations on Manifolds with Multiplicationp. 95
Lie Algebra of Derivationsp. 96
Involutive Automorphism of the Derivation Algebra of a Symmetric Spacep. 97
Symmetric Algebras and Lie Ternariesp. 98
Lie Ternary of a Symmetric Spacep. 100
Cartan Theoremp. 101
Functor Sp. 101
Comparison of the Functor S with the Lie Functor [p. 103
Properties of the Functor Sp. 104
Computation of the Lie Ternary of the Space (G/H)¿-p. 105
Fundamental Group of the Quotient Spacep. 107
Symmetric Space with a Given Lie Ternaryp. 109
Coveringsp. 109
Cartan Theoremp. 110
Identification of Homogeneous Spaces with Quotient Spacesp. 111
Translations of a Symmetric Spacep. 112
Proof of the Cartan Theoremp. 112
Palais and Kobayashi Theoremsp. 114
Infinite-Dimensional Manifolds and Lie Groupsp. 114
Vector Fields Induced by a Lie Group Actionp. 115
Palais Theoremp. 117
Kobayashi Theoremp. 124
Affine Automorphism Groupp. 125
Automorphism Group of a Symmetric Spacep. 125
Translation Group of a Symmetric Spacep. 126
Lagrangians in Riemannian Spacesp. 127
Riemannian and Pseudo-Riemannian Spacesp. 127
Riemannian Connectionsp. 129
Geodesics in a Riemannian Spacep. 133
Simplest Problem of the Calculus of Variationsp. 134
Euler-Lagrange Equationsp. 135
Minimum Curves and Extremalsp. 137
Regular Lagrangiansp. 139
Extremals of the Energy Lagrangianp. 139
Metric Properties of Geodesicsp. 141
Length of a Curve in a Riemannian Spacep. 141
Natural Parameterp. 142
Riemannian Distance and Shortest Arcsp. 142
Extremals of the Length Lagrangianp. 143
Riemannian Coordinatesp. 144
Gauss Lemmap. 145
Geodesics are Locally Shortest Arcsp. 148
Smoothness of Shortest Arcsp. 149
Local Existence of Shortest Arcsp. 150
Intrinsic Metricp. 151
Hopf-Rinow Theoremp. 153
Harmonic Functionals and Related Topicsp. 159
Riemannian Volume Elementp. 159
Discriminant Tensorp. 159
Foss-Weyl Formulap. 160
Case n = 2p. 162
Laplace Operator on a Riemannian Spacep. 164
The Green Formulasp. 165
Existence of Harmonic Functions with a Nonzero Differentialp. 166
Conjugate Harmonic Functionsp. 170
Isothermal Coordinatesp. 172
Semi-Cartesian Coordinatesp. 173
Cartesian Coordinatesp. 175
Minimal Surfacesp. 176
Conformal Coordinatesp. 176
Conformal Structuresp. 177
Minimal Surfacesp. 178
Explanation of Their Namep. 181
Plateau Problemp. 181
Free Relativistic Stringsp. 182
Simplest Problem of the Calculus of Variations for Functions of Two Variablesp. 184
Extremals of the Area Functionalp. 186
Case n = 3p. 188
Representation of Minimal Surfaces Via Holomorphic Functionsp. 189
Weierstrass Formulasp. 190
Adjoined Minimal Surfacesp. 191
Curvature in Riemannian Spacep. 193
Riemannian Curvature Tensorp. 193
Symmetries of the Riemannian Tensorp. 193
Riemannian Tensor as a Functionalp. 198
Walker Identity and Its Consequencesp. 199
Recurrent Spacesp. 200
Virtual Curvature Tensorsp. 201
Reconstruction of the Bianci Tensor from Its Values on Bivectorsp. 202
Sectional Curvaturesp. 204
Formula for the Sectional Curvaturep. 205
Gaussian Curvaturep. 207
Bianchi Tensors as Operatorsp. 207
Splitting of Trace-Free Tensorsp. 208
Gaussian Curvature and the Scalar Curvaturep. 209
Curvature Tensor for n = 2p. 210
Geometric Interpretation of the Sectional Curvaturep. 210
Total Curvature of a Domain on a Surfacep. 212
Rotation of a Vector Field on a Curvep. 214
Rotation of the Field of Tangent Vectorsp. 215
Gauss-Bonnet Formulap. 218
Triangulated Surfacesp. 220
Gauss-Bonnet Theoremp. 221
Some Special Tensorsp. 223
Characteristic Numbersp. 223
Euler Characteristic Numberp. 223
Hodge Operatorp. 225
Euler Number of a 4m-Dimensional Manifoldp. 226
Euler Characteristic of a Manifold of an Arbitrary Dimensionp. 228
Signature Theoremp. 229
Ricci Tensor of a Riemannian Spacep. 230
Ricci Tensor of a Bianchi Tensorp. 231
Einstein and Weyl Tensorsp. 232
Case n = 3p. 234
Einstein Spacesp. 234
Thomas Criterionp. 236
Surfaces with Conformal Structurep. 238
Conformal Transformations of a Metricp. 238
Conformal Curvature Tensorp. 240
Conformal Equivalenciesp. 241
Conformally Flat Spacesp. 242
Conformally Equivalent Surfacesp. 243
Classification of Surfaces with a Conformal Structurep. 243
Surfaces of Parabolic Typep. 244
Surfaces of Elliptic Typep. 245
Surfaces of Hyperbolic Typep. 246
Mappings and Submanifolds Ip. 248
Locally Isometric Mapping of Riemannian Spacesp. 248
Metric Coveringsp. 249
Theorem on Expanding Mappingsp. 250
Isometric Mappings of Riemannian Spacesp. 251
Isometry Group of a Riemannian Spacep. 252
Elliptic Geometryp. 252
Proof of Proposition 18.1p. 253
Dimension of the Isometry Groupp. 253
Killing Fieldsp. 254
Riemannian Connection on a Submanifold of a Riemannian Spacep. 255
Gauss and Weingarten Formulas for Submanifolds of Riemannian Spacesp. 257
Normal of the Mean Curvaturep. 258
Gauss, Peterson-Codazzi, and Ricci Relationsp. 259
Case of a Flat Ambient Spacep. 260
Submanifolds IIp. 262
Locally Symmetric Submanifoldsp. 262
Compact Submanifoldsp. 267
Chern-Kuiper Theoremp. 268
First and Second Quadratic Forms of a Hypersurfacep. 269
Hypersurfaces Whose Points are All Umbilicalp. 271
Principal Curvatures of a Hypersurfacep. 272
Scalar Curvature of a Hypersurfacep. 273
Hypersurfaces That are Einstein Spacesp. 274
Rigidity of the Spherep. 275
Fundamental Forms of a Hypersurfacep. 276
Sufficient Condition for Rigidity of Hypersurfacesp. 276
Hypersurfaces with a Given Second Fundamental Formp. 277
Hypersurfaces with Given First and Second Fundamental Formsp. 278
Proof of the Uniquenessp. 280
Proof of the Existencep. 281
Proof of a Local Variant of the Existence and Uniqueness Theoremp. 282
Spaces of Constant Curvaturep. 288
Spaces of Constant Curvaturep. 288
Model Spaces of Constant Curvaturep. 290
Model Spaces as Hypersurfacesp. 292
Isometries of Model Spacesp. 294
Fixed Points of Isometriesp. 296
Riemann Theoremp. 296
Space Formsp. 298
Space Formsp. 298
Cartan-Killing Theoremp. 299
(Pseudo-)Riemannian Symmetric Spacesp. 299
Classification of Space Formsp. 300
Spherical Forms of Even Dimensionp. 301
Orientable Space Formsp. 302
Complex-Analytic and Conformal Quotient Manifoldsp. 304
Riemannian Spaces with an Isometry Group of Maximal Dimension.p. 304
Their Enumerationp. 306
Complete Mobility Conditionp. 307
Four-Dimensional Manifoldsp. 308
Bianchi Tensors for n = 4p. 308
Matrix Representation of Bianchi Tensors for n = 4p. 309
Explicit Form of Bianchi Tensors for n = 4p. 311
Euler Numbers for n = 4p. 313
Chern-Milnor Theoremp. 314
Sectional Curvatures of Four-Dimensional Einstein Spacesp. 316
Berger Theoremp. 316
Pontryagin Number of a Four-Dimensional Riemannian Spacep. 317
Thorp Theoremp. 319
Sentenac Theoremp. 320
Metrics on a Lie Group Ip. 324
Left-Invariant Metrics on a Lie Groupp. 324
Invariant Metrics on a Lie Groupp. 324
Semisimple Lie Groups and Algebrasp. 326
Simple Lie Groups and Algebrasp. 329
Inner Derivations of Lie Algebrasp. 329
Adjoint Groupp. 331
Lie Groups and Algebras Without Centerp. 332
Metrics on a Lie Group IIp. 333
Maurer-Cartan Formsp. 333
Left-Invariant Differential Formsp. 334
Haar Measure on a Lie groupp. 336
Unimodular Lie Groupsp. 339
Invariant Riemannian Metrics on a Compact Lie Groupp. 340
Lie Groups with a Compact Lie Algebrap. 341
Weyl Theoremp. 343
Jacobi Theoryp. 344
Conjugate Pointsp. 344
Second Variation of Lengthp. 345
Formula for the Second Variationp. 346
Reduction of the Problemp. 348
Minimal Fields and Jacobi Fieldsp. 349
Jacobi Variationp. 351
Jacobi Fields and Conjugate Pointsp. 353
Properties of Jacobi Fieldsp. 353
Minimality of Normal Jacobi Fieldsp. 355
Proof of the Jacobi Theoremp. 358
Some Additional Theorems Ip. 360
Cut Pointsp. 360
Lemma on Continuityp. 361
Cut Loci and Maximal Normal Neighborhoodsp. 362
Proof of Lemma 28.1p. 364
Spaces of Strictly Positive Ricci Curvaturep. 367
Mayers Theoremp. 368
Spaces of Strictly Positive Sectional Curvaturep. 369
Spaces of Nonpositive Sectional Curvaturep. 370
Chapter 29. Some Additional Theorems IIp. 371
Cartan-Hadamard Theoremp. 371
Consequence of the Cartan-Hadamard Theoremp. 374
Cartan-Killing Theorem for K = 0p. 375
Bochner Theoremp. 375
Operators AXp. 376
Infinitesimal Variant of the Bochner Theoremp. 378
Isometry Group of a Compact Spacep. 378
Addendump. 381
Smooth Manifoldsp. 381
Introductory Remarksp. 381
Open Sets in the Space Rn and Their Diffeomorphismsp. 381
Charts and Atlasesp. 383
Maximal Atlasesp. 385
Smooth Manifoldsp. 386
Smooth Manifold Topologyp. 386
Smooth Structures on a Topological Spacep. 390
DIFF Categoryp. 391
Transport of Smooth Structuresp. 392
Tangent Vectorsp. 394
Vectors Tangent to a Smooth Manifoldp. 394
Oriented Manifoldsp. 396
Differential of a Smooth Mappingp. 397
Chain Rulep. 398
Gradient of a Smooth Functionp. 399
Etale Mapping Theoremp. 400
Theorem on a Local Coordinate Changep. 400
Locally Flat Mappingsp. 401
Immersions and Submersionsp. 402
Submanifolds of a Smooth Manifoldp. 404
Submanifolds of a Smooth Manifoldp. 404
Subspace Tangent to a Submanifoldp. 405
Local Representation of a Submanifoldp. 405
Uniqueness of a Submanifold Structurep. 407
Case of Embedded Submanifoldsp. 407
Tangent Space of a Direct Productp. 408
Theorem on the Inverse Image of a Regular Valuep. 409
Solution of Sets of Equationsp. 410
Embedding Theoremp. 411
Vector and Tensor Fields. Differential Formsp. 413
Tensor Fieldsp. 413
Vector Fields and Derivationsp. 416
Lie Algebra of Vector Fieldsp. 419
Integral Curves of Vector Fieldsp. 421
Vector Fields and Flowsp. 422
Transport of Vector Fields via Diffeomorphismsp. 423
Lie Derivative of a Tensor Fieldp. 425
Linear Differential Formsp. 426
Differential Forms of an Arbitrary Degreep. 428
Differential Forms as Functionals on Vector Fieldsp. 429
Inner Product of Vector Fields and Differential Formsp. 430
Transport of a Differential Form via a Smooth Mappingp. 431
Exterior Differentialp. 433
Vector Bundlesp. 436
Bundles and Their Morphismsp. 436
Vector Bundlesp. 438
Sections of Vector Bundlesp. 439
Morphisms of Vector Bundlesp. 440
Trivial Vector Bundlesp. 442
Tangent Bundlesp. 442
Frame Bundlesp. 445
Metricizable Bundlesp. 447
-Tensor Fieldsp. 447
Multilinear Functions and-Tensor Fieldsp. 449
Tensor Product of Vector Bundlesp. 450
Generalizationp. 450
Tensor Product of Sectionsp. 451
Inverse Image of a Vector Bundlep. 452
Connections on Vector Bundlesp. 454
Vertical Subspacesp. 454
Fields of Horizontal Subspacesp. 455
Connections and Their Formsp. 457
Inverse Image of a Connectionp. 459
Horizontal Curvesp. 460
Covariant Derivatives of Sectionsp. 461
Covariant Differentiations Along a Curvep. 463
Connections as Covariant Differentiationsp. 463
Connections on Metricized Bundlesp. 465
Covariant Differentialp. 465
Comparison of Various Definitions of Connectionp. 469
Connections on Frame Bundlesp. 470
Comparison with Connections on Vector Bundlesp. 473
Curvature Tensorp. 475
Parallel Translation Along a Curvep. 475
Computation of the Parallel Translation Along a Loopp. 477
Curvature Operator at a Given Pointp. 482
Translation of a Vector Along an Infinitely Small Parallelogramp. 484
Curvature Tensorp. 485
Formula for Transforming Coordinates of the Curvature Tensorp. 486
Expressing the Curvature Operator via Covariant Derivativesp. 487
Cartan Structural Equationp. 490
Bianchi Identityp. 491
Suggested Readingp. 494
Indexp. 495
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540411086
ISBN-10: 3540411089
Series: Encyclopaedia of Mathematical Sciences
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 504
Published: April 2001
Publisher: SPRINGER VERLAG GMBH
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 2.87
Weight (kg): 0.91

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