Lectures On Differential Geometry (University Mathematics)

✍ Scribed by S S Chern, W. H. Chen, K. S. Lam

• Publisher: Wspc
• Year: 2000
• Tongue: English
• Leaves: 367
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Contents
Chapter 1 Differentiable Manifolds
§1-1 Definition of Differentiable Manifolds
§1-2 Tangent Spaces
§1-3 Submanifolds
§1-4 Frobenius’ Theorem
Chapter 2 Multilinear Algebra
§2-1 Tensor Products
§2-2 Tensors
§2-3 Exterior Algebra
Chapter 3 Exterior Differential Calculus
§3-1 Tensor Bundles and Vector Bundles
§3-2 Exterior Differentiation
§3-3 Integrals of Differential Forms
§3-4 Stokes’ Formula
Chapter 4 Connections
§4-1 Connections on Vector Bundles
§4-2 Affine Connections
§4-3 Connections on Frame Bundles
Chapter 5 Riemannian Geometry
§5-1 The Fundamental Theorem of Riemannian Geometry
§5-2 Geodesic Normal Coordinates
§5-3 Sectional Curvature
§5-4 The Gauss-Bonnet Theorem
Chapter 6 Lie Groups and Moving Frames
§6-1 Lie Groups
§6-2 Lie Transformation Groups
§6-3 The Method of Moving Frames
§6-4 Theory of Surfaces
Chapter 7 Complex Manifolds
§7-1 Complex Manifolds
§7-2 The Complex Structure on a Vector Space
§7-3 Almost Complex Manifolds
§7-4 Connections on Complex Vector Bundles
§7-5 Hermitian Manifolds and Kählerian Manifolds
Chapter 8 Finsler Geometry
§8-1 Preliminaries
§8-2 Geometry on the Projectivised Tangent Bundle (PTM) and the Hilbert Form
§8-3 The Chern Connection
§8-3.1 Determination of the Connection
§8-3.2 The Cartan Tensor and Characterization of Riemannian Geometry
§8-3.3 Explicit Formulas for the Connection Forms in Natural Coordinates
§8-4 Structure Equations and the Flag Curvature
$8-4.1 The Curvature Tensor
§8-4.2 The Flag Curvature and the Ricci Curvature
§8-4.3 Special Finsler Spaces
§8-5 The First Variation of Arc Length and Geodesics
§8-6 The Second Variation of Arc Length and Jacobi Fields
§8-7 Completeness and the Hopf-Rinow Theorem
§8-8 The Theorems of Bonnet-Myers and Synge
Appendix A Historical Notes
§A-1 Classical Differential Geometry
§A-2 Riemannian Geometry
§A-3 Manifolds
§A-4 Global Geometry
Appendix B Differential Geometry and Theoretical Physics
§B-1 Dynamics and Moving Frames
§B-2 Theory of Surfaces, Solitons and the Sigma Model
§B-3 Gauge Field Theory
§B-4 Conclusion
References
Index