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