This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives.
Author(s): Shiing-Shen Chern, Wei-Huan Chen, K. S. Lam
Series: Series on University Mathematics, Volume 1
Publisher: World Scientific
Year: 2000
Language: English
Pages: 368
Tags: Математика;Топология;Дифференциальная геометрия и топология;Дифференциальная геометрия;
Preface......Page 6
Contents......Page 10
§1-1 Definition of Differentiable Manifolds......Page 12
§1-2 Tangent Spaces......Page 20
§1-3 Submanifolds......Page 29
§1-4 Frobenius’ Theorem......Page 40
§2-1 Tensor Products......Page 50
§2-2 Tensors......Page 58
§2-3 Exterior Algebra......Page 63
§3-1 Tensor Bundles and Vector Bundles......Page 76
§3-2 Exterior Differentiation......Page 85
§3-3 Integrals of Differential Forms......Page 96
§3-4 Stokes’ Formula......Page 103
§4-1 Connections on Vector Bundles......Page 112
§4-2 Affine Connections......Page 124
§4-3 Connections on Frame Bundles......Page 132
§5-1 The Fundamental Theorem of Riemannian Geometry......Page 144
§5-2 Geodesic Normal Coordinates......Page 154
§5-3 Sectional Curvature......Page 166
§5-4 The Gauss-Bonnet Theorem......Page 173
§6-1 Lie Groups......Page 184
§6-2 Lie Transformation Groups......Page 197
§6-3 The Method of Moving Frames......Page 209
§6-4 Theory of Surfaces......Page 221
§7-1 Complex Manifolds......Page 232
§7-2 The Complex Structure on a Vector Space......Page 238
§7-3 Almost Complex Manifolds......Page 247
§7-4 Connections on Complex Vector Bundles......Page 255
§7-5 Hermitian Manifolds and Kählerian Manifolds......Page 267
§8-1 Preliminaries......Page 276
§8-2 Geometry on the Projectivised Tangent Bundle (PTM) and the Hilbert Form......Page 278
§8-3 The Chern Connection......Page 284
§8-3.1 Determination of the Connection......Page 285
§8-3.2 The Cartan Tensor and Characterization of Riemannian Geometry......Page 291
§8-3.3 Explicit Formulas for the Connection Forms in Natural Coordinates......Page 294
§8-4 Structure Equations and the Flag Curvature......Page 299
$8-4.1 The Curvature Tensor......Page 300
§8-4.2 The Flag Curvature and the Ricci Curvature......Page 304
§8-4.3 Special Finsler Spaces......Page 306
§8-5 The First Variation of Arc Length and Geodesics......Page 308
§8-6 The Second Variation of Arc Length and Jacobi Fields......Page 317
§8-7 Completeness and the Hopf-Rinow Theorem......Page 325
§8-8 The Theorems of Bonnet-Myers and Synge......Page 336
§A-2 Riemannian Geometry......Page 342
§A-4 Global Geometry......Page 343
Appendix B Differential Geometry and Theoretical Physics......Page 346
§B-1 Dynamics and Moving Frames......Page 347
§B-2 Theory of Surfaces, Solitons and the Sigma Model......Page 349
§B-3 Gauge Field Theory......Page 351
§B-4 Conclusion......Page 352
References......Page 354
Index......Page 358
