Geometry VI : Riemannian Geometry

Riemannian Geometry

By: M. M. Postnikov, S. A. Vakhrameev (Translator)

Write A Review

Hardcover | 1 April 2001

At a Glance

Hardcover
526 Pages

Dimensions(cm)
23.5 x 15.88 x 3.18

Hardcover

$219.75

or 4 interest-free payments of $54.94 with

Ships in 5 to 7 business days

The original Russian edition of this book is the fifth in my series "Lectures on Geometry. " Therefore, to make the presentation relatively independent and self-contained in the English translation, I have added supplementary chapters in a special addendum (Chaps. 3Q-36), in which the necessary facts from manifold theory and vector bundle theory are briefly summarized without proofs as a rule. In the original edition, the book is divided not into chapters but into lec- tures. This is explained by its origin as classroom lectures that I gave. The principal distinction between chapters and lectures is that the material of each chapter should be complete to a certain extent and the length of chapters can differ, while, in contrast, all lectures should be approximately the same in length and the topic of any lecture can change suddenly in the middle. For the series "Encyclopedia of Mathematical Sciences," the origin of a book has no significance, and the name "chapter" is more usual. Therefore, the name of subdivisions was changed in the translation, although no structural surgery was performed. I have also added a brief bibliography, which was absent in the original edition. The first ten chapters are devoted to the geometry of affine connection spaces. In the first chapter, I present the main properties of geodesics in these spaces. Chapter 2 is devoted to the formalism of covariant derivatives, torsion tensor, and curvature tensor. The major part of Chap.

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)

Shipping

	Standard Shipping	Express Shipping
Metro postcodes:	$9.99	$14.95
Regional postcodes:	$9.99	$14.95
Rural postcodes:	$9.99	$14.95

Orders over $79.00 qualify for free shipping.

How to return your order

At Booktopia, we offer hassle-free returns in accordance with our returns policy. If you wish to return an item, please get in touch with Booktopia Customer Care.

Additional postage charges may be applicable.

Defective items

If there is a problem with any of the items received for your order then the Booktopia Customer Care team is ready to assist you.

For more info please visit our Help Centre.

You Can Find This Book In

Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Medicine Medicine in General Education

Preface	p. V
Affine Connections	p. 1
Connection on a Manifold	p. 1
Covariant Differentiation and Parallel Translation Along a Curve	p. 3
Geodesics	p. 4
Exponential Mapping and Normal Neighborhoods	p. 7
Whitehead Theorem	p. 9
Normal Convex Neighborhoods	p. 13
Existence of Leray Coverings	p. 13
Covariant Differentiation. Curvature	p. 14
Covariant Differentiation	p. 14
The Case of Tensors of Type (r, 1)	p. 16
Torsion Tensor and Symmetric Connections	p. 18
Geometric Meaning of the Symmetry of a Connection	p. 20
Commutativity of Second Covariant Derivatives	p. 21
Curvature Tensor of an Affine Connection	p. 22
Space with Absolute Parallelism	p. 24
Bianci Identities	p. 24
Trace of the Curvature Tensor	p. 27
Ricci Tensor	p. 27
Affine Mappings. Submanifolds	p. 29
Affine Mappings	p. 29
Affinities	p. 32
Affine Coverings	p. 33
Restriction of a Connection to a Submanifold	p. 35
Induced Connection on a Normalized Submanifold	p. 37
Gauss Formula and the Second Fundamental Form of a Normalized Submanifold	p. 38
Totally Geodesic and Auto-Parallel Submanifolds	p. 40
Normal Connection and the Weingarten Formula	p. 42
Van der Waerden-Bortolotti Connection	p. 42
Structural Equations. Local Symmetries	p. 44
Torsion and Curvature Forms	p. 44
Cartan Structural Equations in Polar Coordinates	p. 47
Existence of Affine Local Mappings	p. 50
Locally Symmetric Affine Connection Spaces	p. 51
Local Geodesic Symmetries	p. 53
Semisymmetric Spaces	p. 54
Symmetric Spaces	p. 55
Globally Symmetric Spaces	p. 55
Germs of Smooth Mappings	p. 55
Extensions of Affine Mappings	p. 56
Uniqueness Theorem	p. 58
Reduction of Locally Symmetric Spaces to Globally Symmetric Spaces	p. 59
Properties of Symmetries in Globally Symmetric Spaces	p. 60
Symmetric Spaces	p. 61
Examples of Symmetric Spaces	p. 62
Coincidence of Classes of Symmetric and Globally Symmetric Spaces	p. 63
Connections on Lie Groups	p. 67
Invariant Construction of the Canonical Connection	p. 67
Morphisms of Symmetric Spaces as Affine Mappings	p. 69
Left-Invariant Connections on a Lie Group	p. 70
Cartan Connections	p. 71
Left Cartan Connection	p. 73
Right-Invariant Vector Fields	p. 74
Right Cartan Connection	p. 76
Lie Functor	p. 77
Categories	p. 77
Functors	p. 78
Lie Functor	p. 79
Kernel and Image of a Lie Group Homomorphism	p. 80
Campbell-Hausdorff Theorem	p. 82
Dynkin Polynomials	p. 83
Local Lie Groups	p. 84
Bijectivity of the Lie Functor	p. 85
Affine Fields and Related Topics	p. 87
Affine Fields	p. 87
Dimension of the Lie Algebra of Affine Fields	p. 89
Completeness of Affine Fields	p. 91
Mappings of Left and Right Translation on a Symmetric Space	p. 94
Derivations on Manifolds with Multiplication	p. 95
Lie Algebra of Derivations	p. 96
Involutive Automorphism of the Derivation Algebra of a Symmetric Space	p. 97
Symmetric Algebras and Lie Ternaries	p. 98
Lie Ternary of a Symmetric Space	p. 100
Cartan Theorem	p. 101
Functor S	p. 101
Comparison of the Functor S with the Lie Functor [	p. 103
Properties of the Functor S	p. 104
Computation of the Lie Ternary of the Space (G/H)¿-	p. 105
Fundamental Group of the Quotient Space	p. 107
Symmetric Space with a Given Lie Ternary	p. 109
Coverings	p. 109
Cartan Theorem	p. 110
Identification of Homogeneous Spaces with Quotient Spaces	p. 111
Translations of a Symmetric Space	p. 112
Proof of the Cartan Theorem	p. 112
Palais and Kobayashi Theorems	p. 114
Infinite-Dimensional Manifolds and Lie Groups	p. 114
Vector Fields Induced by a Lie Group Action	p. 115
Palais Theorem	p. 117
Kobayashi Theorem	p. 124
Affine Automorphism Group	p. 125
Automorphism Group of a Symmetric Space	p. 125
Translation Group of a Symmetric Space	p. 126
Lagrangians in Riemannian Spaces	p. 127
Riemannian and Pseudo-Riemannian Spaces	p. 127
Riemannian Connections	p. 129
Geodesics in a Riemannian Space	p. 133
Simplest Problem of the Calculus of Variations	p. 134
Euler-Lagrange Equations	p. 135
Minimum Curves and Extremals	p. 137
Regular Lagrangians	p. 139
Extremals of the Energy Lagrangian	p. 139
Metric Properties of Geodesics	p. 141
Length of a Curve in a Riemannian Space	p. 141
Natural Parameter	p. 142
Riemannian Distance and Shortest Arcs	p. 142
Extremals of the Length Lagrangian	p. 143
Riemannian Coordinates	p. 144
Gauss Lemma	p. 145
Geodesics are Locally Shortest Arcs	p. 148
Smoothness of Shortest Arcs	p. 149
Local Existence of Shortest Arcs	p. 150
Intrinsic Metric	p. 151
Hopf-Rinow Theorem	p. 153
Harmonic Functionals and Related Topics	p. 159
Riemannian Volume Element	p. 159
Discriminant Tensor	p. 159
Foss-Weyl Formula	p. 160
Case n = 2	p. 162
Laplace Operator on a Riemannian Space	p. 164
The Green Formulas	p. 165
Existence of Harmonic Functions with a Nonzero Differential	p. 166
Conjugate Harmonic Functions	p. 170
Isothermal Coordinates	p. 172
Semi-Cartesian Coordinates	p. 173
Cartesian Coordinates	p. 175
Minimal Surfaces	p. 176
Conformal Coordinates	p. 176
Conformal Structures	p. 177
Minimal Surfaces	p. 178
Explanation of Their Name	p. 181
Plateau Problem	p. 181
Free Relativistic Strings	p. 182
Simplest Problem of the Calculus of Variations for Functions of Two Variables	p. 184
Extremals of the Area Functional	p. 186
Case n = 3	p. 188
Representation of Minimal Surfaces Via Holomorphic Functions	p. 189
Weierstrass Formulas	p. 190
Adjoined Minimal Surfaces	p. 191
Curvature in Riemannian Space	p. 193
Riemannian Curvature Tensor	p. 193
Symmetries of the Riemannian Tensor	p. 193
Riemannian Tensor as a Functional	p. 198
Walker Identity and Its Consequences	p. 199
Recurrent Spaces	p. 200
Virtual Curvature Tensors	p. 201
Reconstruction of the Bianci Tensor from Its Values on Bivectors	p. 202
Sectional Curvatures	p. 204
Formula for the Sectional Curvature	p. 205
Gaussian Curvature	p. 207
Bianchi Tensors as Operators	p. 207
Splitting of Trace-Free Tensors	p. 208
Gaussian Curvature and the Scalar Curvature	p. 209
Curvature Tensor for n = 2	p. 210
Geometric Interpretation of the Sectional Curvature	p. 210
Total Curvature of a Domain on a Surface	p. 212
Rotation of a Vector Field on a Curve	p. 214
Rotation of the Field of Tangent Vectors	p. 215
Gauss-Bonnet Formula	p. 218
Triangulated Surfaces	p. 220
Gauss-Bonnet Theorem	p. 221
Some Special Tensors	p. 223
Characteristic Numbers	p. 223
Euler Characteristic Number	p. 223
Hodge Operator	p. 225
Euler Number of a 4m-Dimensional Manifold	p. 226
Euler Characteristic of a Manifold of an Arbitrary Dimension	p. 228
Signature Theorem	p. 229
Ricci Tensor of a Riemannian Space	p. 230
Ricci Tensor of a Bianchi Tensor	p. 231
Einstein and Weyl Tensors	p. 232
Case n = 3	p. 234
Einstein Spaces	p. 234
Thomas Criterion	p. 236
Surfaces with Conformal Structure	p. 238
Conformal Transformations of a Metric	p. 238
Conformal Curvature Tensor	p. 240
Conformal Equivalencies	p. 241
Conformally Flat Spaces	p. 242
Conformally Equivalent Surfaces	p. 243
Classification of Surfaces with a Conformal Structure	p. 243
Surfaces of Parabolic Type	p. 244
Surfaces of Elliptic Type	p. 245
Surfaces of Hyperbolic Type	p. 246
Mappings and Submanifolds I	p. 248
Locally Isometric Mapping of Riemannian Spaces	p. 248
Metric Coverings	p. 249
Theorem on Expanding Mappings	p. 250
Isometric Mappings of Riemannian Spaces	p. 251
Isometry Group of a Riemannian Space	p. 252
Elliptic Geometry	p. 252
Proof of Proposition 18.1	p. 253
Dimension of the Isometry Group	p. 253
Killing Fields	p. 254
Riemannian Connection on a Submanifold of a Riemannian Space	p. 255
Gauss and Weingarten Formulas for Submanifolds of Riemannian Spaces	p. 257
Normal of the Mean Curvature	p. 258
Gauss, Peterson-Codazzi, and Ricci Relations	p. 259
Case of a Flat Ambient Space	p. 260
Submanifolds II	p. 262
Locally Symmetric Submanifolds	p. 262
Compact Submanifolds	p. 267
Chern-Kuiper Theorem	p. 268
First and Second Quadratic Forms of a Hypersurface	p. 269
Hypersurfaces Whose Points are All Umbilical	p. 271
Principal Curvatures of a Hypersurface	p. 272
Scalar Curvature of a Hypersurface	p. 273
Hypersurfaces That are Einstein Spaces	p. 274
Rigidity of the Sphere	p. 275
Fundamental Forms of a Hypersurface	p. 276
Sufficient Condition for Rigidity of Hypersurfaces	p. 276
Hypersurfaces with a Given Second Fundamental Form	p. 277
Hypersurfaces with Given First and Second Fundamental Forms	p. 278
Proof of the Uniqueness	p. 280
Proof of the Existence	p. 281
Proof of a Local Variant of the Existence and Uniqueness Theorem	p. 282
Spaces of Constant Curvature	p. 288
Spaces of Constant Curvature	p. 288
Model Spaces of Constant Curvature	p. 290
Model Spaces as Hypersurfaces	p. 292
Isometries of Model Spaces	p. 294
Fixed Points of Isometries	p. 296
Riemann Theorem	p. 296
Space Forms	p. 298
Space Forms	p. 298
Cartan-Killing Theorem	p. 299
(Pseudo-)Riemannian Symmetric Spaces	p. 299
Classification of Space Forms	p. 300
Spherical Forms of Even Dimension	p. 301
Orientable Space Forms	p. 302
Complex-Analytic and Conformal Quotient Manifolds	p. 304
Riemannian Spaces with an Isometry Group of Maximal Dimension.	p. 304
Their Enumeration	p. 306
Complete Mobility Condition	p. 307
Four-Dimensional Manifolds	p. 308
Bianchi Tensors for n = 4	p. 308
Matrix Representation of Bianchi Tensors for n = 4	p. 309
Explicit Form of Bianchi Tensors for n = 4	p. 311
Euler Numbers for n = 4	p. 313
Chern-Milnor Theorem	p. 314
Sectional Curvatures of Four-Dimensional Einstein Spaces	p. 316
Berger Theorem	p. 316
Pontryagin Number of a Four-Dimensional Riemannian Space	p. 317
Thorp Theorem	p. 319
Sentenac Theorem	p. 320
Metrics on a Lie Group I	p. 324
Left-Invariant Metrics on a Lie Group	p. 324
Invariant Metrics on a Lie Group	p. 324
Semisimple Lie Groups and Algebras	p. 326
Simple Lie Groups and Algebras	p. 329
Inner Derivations of Lie Algebras	p. 329
Adjoint Group	p. 331
Lie Groups and Algebras Without Center	p. 332
Metrics on a Lie Group II	p. 333
Maurer-Cartan Forms	p. 333
Left-Invariant Differential Forms	p. 334
Haar Measure on a Lie group	p. 336
Unimodular Lie Groups	p. 339
Invariant Riemannian Metrics on a Compact Lie Group	p. 340
Lie Groups with a Compact Lie Algebra	p. 341
Weyl Theorem	p. 343
Jacobi Theory	p. 344
Conjugate Points	p. 344
Second Variation of Length	p. 345
Formula for the Second Variation	p. 346
Reduction of the Problem	p. 348
Minimal Fields and Jacobi Fields	p. 349
Jacobi Variation	p. 351
Jacobi Fields and Conjugate Points	p. 353
Properties of Jacobi Fields	p. 353
Minimality of Normal Jacobi Fields	p. 355
Proof of the Jacobi Theorem	p. 358
Some Additional Theorems I	p. 360
Cut Points	p. 360
Lemma on Continuity	p. 361
Cut Loci and Maximal Normal Neighborhoods	p. 362
Proof of Lemma 28.1	p. 364
Spaces of Strictly Positive Ricci Curvature	p. 367
Mayers Theorem	p. 368
Spaces of Strictly Positive Sectional Curvature	p. 369
Spaces of Nonpositive Sectional Curvature	p. 370
Chapter 29. Some Additional Theorems II	p. 371
Cartan-Hadamard Theorem	p. 371
Consequence of the Cartan-Hadamard Theorem	p. 374
Cartan-Killing Theorem for K = 0	p. 375
Bochner Theorem	p. 375
Operators AX	p. 376
Infinitesimal Variant of the Bochner Theorem	p. 378
Isometry Group of a Compact Space	p. 378
Addendum	p. 381
Smooth Manifolds	p. 381
Introductory Remarks	p. 381
Open Sets in the Space Rn and Their Diffeomorphisms	p. 381
Charts and Atlases	p. 383
Maximal Atlases	p. 385
Smooth Manifolds	p. 386
Smooth Manifold Topology	p. 386
Smooth Structures on a Topological Space	p. 390
DIFF Category	p. 391
Transport of Smooth Structures	p. 392
Tangent Vectors	p. 394
Vectors Tangent to a Smooth Manifold	p. 394
Oriented Manifolds	p. 396
Differential of a Smooth Mapping	p. 397
Chain Rule	p. 398
Gradient of a Smooth Function	p. 399
Etale Mapping Theorem	p. 400
Theorem on a Local Coordinate Change	p. 400
Locally Flat Mappings	p. 401
Immersions and Submersions	p. 402
Submanifolds of a Smooth Manifold	p. 404
Submanifolds of a Smooth Manifold	p. 404
Subspace Tangent to a Submanifold	p. 405
Local Representation of a Submanifold	p. 405
Uniqueness of a Submanifold Structure	p. 407
Case of Embedded Submanifolds	p. 407
Tangent Space of a Direct Product	p. 408
Theorem on the Inverse Image of a Regular Value	p. 409
Solution of Sets of Equations	p. 410
Embedding Theorem	p. 411
Vector and Tensor Fields. Differential Forms	p. 413
Tensor Fields	p. 413
Vector Fields and Derivations	p. 416
Lie Algebra of Vector Fields	p. 419
Integral Curves of Vector Fields	p. 421
Vector Fields and Flows	p. 422
Transport of Vector Fields via Diffeomorphisms	p. 423
Lie Derivative of a Tensor Field	p. 425
Linear Differential Forms	p. 426
Differential Forms of an Arbitrary Degree	p. 428
Differential Forms as Functionals on Vector Fields	p. 429
Inner Product of Vector Fields and Differential Forms	p. 430
Transport of a Differential Form via a Smooth Mapping	p. 431
Exterior Differential	p. 433
Vector Bundles	p. 436
Bundles and Their Morphisms	p. 436
Vector Bundles	p. 438
Sections of Vector Bundles	p. 439
Morphisms of Vector Bundles	p. 440
Trivial Vector Bundles	p. 442
Tangent Bundles	p. 442
Frame Bundles	p. 445
Metricizable Bundles	p. 447
-Tensor Fields	p. 447
Multilinear Functions and-Tensor Fields	p. 449
Tensor Product of Vector Bundles	p. 450
Generalization	p. 450
Tensor Product of Sections	p. 451
Inverse Image of a Vector Bundle	p. 452
Connections on Vector Bundles	p. 454
Vertical Subspaces	p. 454
Fields of Horizontal Subspaces	p. 455
Connections and Their Forms	p. 457
Inverse Image of a Connection	p. 459
Horizontal Curves	p. 460
Covariant Derivatives of Sections	p. 461
Covariant Differentiations Along a Curve	p. 463
Connections as Covariant Differentiations	p. 463
Connections on Metricized Bundles	p. 465
Covariant Differential	p. 465
Comparison of Various Definitions of Connection	p. 469
Connections on Frame Bundles	p. 470
Comparison with Connections on Vector Bundles	p. 473
Curvature Tensor	p. 475
Parallel Translation Along a Curve	p. 475
Computation of the Parallel Translation Along a Loop	p. 477
Curvature Operator at a Given Point	p. 482
Translation of a Vector Along an Infinitely Small Parallelogram	p. 484
Curvature Tensor	p. 485
Formula for Transforming Coordinates of the Curvature Tensor	p. 486
Expressing the Curvature Operator via Covariant Derivatives	p. 487
Cartan Structural Equation	p. 490
Bianchi Identity	p. 491
Suggested Reading	p. 494
Index	p. 495
Table of Contents provided by Publisher. All Rights Reserved.

Geometry VI : Riemannian Geometry

Riemannian Geometry

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

More in Differential & Riemannian Geometry

An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised

Volume 120

Lectures on Differential Geometry I

Differentiable Manifolds

Geometric Methods in Physical Systems

From Differentiable Structures to Applications, the Wisla 22 Winter School and Workshop

Geometry of Integrable Systems

An Introduction

Consensus and Synchronization

From the Euclidean Space and the Circle to Lie Groups

Collected Papers of Y. Matsushima

SERIES IN PURE MATHEMATICS

Harmonic Maps

Concise Complex Analysis

Topics in Integral Geometry

Series in Pure Mathematics

Two Reports on Harmonic Maps

Introductory Differential Geometry for Physicists

Least Action Principle of Crystal Formation of Dense Packing Type and the Proof of Kepler's Conjecture

Nankai Tracts in Mathematics

Lectures on Chern-Weil Theory and Witten Deformations

Nankai Traits in Mathematics

Web Theory and Related Topics

Trends in Complex Analysis, Differential Geometry and Mathematical Physics

Differential Geometry

Dover Books on Mathema 1.4tics

Lectures on Classical Differential Geometry

Second Edition

High-Dimensional Manifold Topology : Proceedings of the School ICTP, Trieste, Italy, 21 May-8 June 2001

Proceedings of the School ICTP, Trieste, Italy, 21 May-8 June 2001

Total Mean Curvature and Submanifolds of Finite Type

Series In Pure Mathematics

Introduction to Compact Lie Groups

SERIES IN PURE MATHEMATICS

Variational Theory of Geodesics

Dover Books on Mathematics

Geometry of Submanifolds

Dover Books on Mathematics

All the Math You Missed

(But Need to Know for Graduate School)

Visual Differential Geometry and Forms

A Mathematical Drama in Five Acts

This product is categorised by