微分几何讲义LECTURES ON DIFFERENTIAL GEOMETRY

内容概要

Shiing-Shen Chern (October 26, 1911 – December 3, 2004) was a Chinese-born American mathematician. He was regarded as one of the leaders in differential geometry of the twentieth century.

Chern received numerous honors and awards in his life, including:

1970, Chauvenet Prize, of the Mathematical Association of America;

1975, National Medal of Science[5]

1982, Humboldt Prize, Germany;

1983, Leroy P. Steele Prize, of the American Mathematical Society;

1984, Wolf Prize in Mathematics, Israel;

2002, Lobachevsky Medal;

2004 May, Shaw Prize in mathematical sciences, Hong Kong;[6]

1948, Academician, Academia Sinica;

1950, Honorary Member, Indian Mathematical Society;

1961, Member, United States National Academy of Sciences;

1963, Fellow, American Academy of Arts and Sciences;

1971, Corresponding Member, Brazilian Academy of Sciences;

1983, Associate Founding Fellow, TWAS;

1985, Foreign Fellow, Royal Society of London, UK;

1986, Honorary Fellow, London Mathematical Society, UK;

1986, Corresponding Member, Academia Peloritana, Messina, Sicily;

1987, Honorary Life Member, New York Academy of Sciences;

1989, Foreign Member, Accademia dei Lincei, Italy;

1989, Foreign Member, Académie des sciences, France;

1989, Member, American Philosophical Society;

1994, Foreign Member, Chinese Academy of Sciences.

书籍目录

1 Differentiable Manifolds  1-1 Definition of Differentiable Manifolds  1-2 Tangent Spaces  1-3 Submanifolds  1-4 Frobenius' Theorem2 Multilinear Algebra  2-1 Tensor Products  2-2 Tensors  2-3 Exterior Algebra3 Exterior Differential Calculus  3-1 Tensor Bundles and Vector Bundles  3-2 Exterior Differentiation  3-3 Integrals of Differential Forms  3-4 Stokes' Formula4 Connections  4-1 Connections on Vector Bundles  4-2 Affine Connections  4-3 Connections on Frame Bundles5 Riemannian Geometry  5-1 The Fundamental Theorem of Riemannian Geometry . .  5-2 Geodesic Normal Coordinates  5-3 Sectional Curvature  5-4 The Gauss-Bonnet Theorem6 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 Surfaces7 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 K  ihlerian Manifolds8 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 Riemannlan 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 SyngeA Historical Notes  A-1 Classical Differential Geometry  A-2 Riemannian Geometry  A-3Manifolds  A-4 Global GeometryB 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 ConclusionReferencesIndex

作者简介

This is a translation of an 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, sought to combine 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 fibre 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 specifically for this edition by Professor Chern to bring the text into perspective.

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