微分几何讲义LECTURES ON DIFFERENTIAL GEOMETRY

1999-11  

World Scientific Pub Co Inc  

Chern, Shiing-Shen/ Chen, W. H./ Lam, K. S.  

356  

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Professor Shiing-shen Chern retired from UC Berkeley and is now based in the Nankai Institute of Mathematics, which he founded in 1985. He is also the founding director of the Mathematical Science Research Institute, Berkeley (1981). He was awarded the National Science Medal in 1975 and Wolf Prize in Mathematics in 1983/4. His area of research was differential geometry where he studied the (now named) Chern characteristic classes in fibre spaces. The Chern Visiting Professorship, begun in 1996, honors the Berkeley professor emeritus widely regarded as the greatest geometer of his generation. "Chern's belief in young people and his encouragement of them had a lot to do with the spectacular growth of geometry in the second half of this century", mathematician Blaine Lawson has said. "It is not easy to find a geometer who was not for some period of time either a student or a post-doctoral fellow in the orbit of Chern".

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

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