导语

  

内容提要

  

    本书系统地论述了微分几何的基本知识。前3章，以及第6章共计4章介绍了流形、多重线性函数等。第7章是复流形。最后两个附录，介绍了大范围曲线论和曲面论以及微分几何与理论物理关系的论述。

1  Differentiable Manifolds
  §1-1  Definition of Differentiable Manifolds
  §1-2  Tangent Spaces
  §1-3  Submanifolds
  §1-4  Frobenius'Theorem
2  Multilinear Algebra
  §2-1  Tensor Products
  §2-2  Tensors
  §2-3  Exterior Algebra
3  Exterior Differential Calcultus
  §3-1  Tensor Bundles and Vector Bundles
  §3-2  Exterior Differentiation
  §3-3  Integrals of Differential Forms
  §3-4  Stokes'Formula
4  Connections
  §4-1  Connections on Vector Bundles
  §4-2  Affine Connections
  §4-3  Connections on Frame Bundles
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
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
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 Kihlerian Manifolds
8  Finsler Geometry
  §8-1  Preliminaries
  §8-2  Geometry on the Projectivised Tangent Bundie (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
    §8-3.3  Explicit Formulas for the Connection Formg in Naturali
  §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
A  Historical Notes
  §A-1  Classical Differential Geometry
  §A-2  Riemannian Geometry
  §A-3  Manifolds
  §A-4  Global Geometry
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