Preface to the English Edition Preface to the German Edition Chapter 1. Notations and Prerequisites from Analysis Chapter 2. Curves in IRn 2A Frenet curves in IRn 2B Plane curves and space curves 2C Relations between the curvature and the torsion 2D The Frenet equations and the fundamental theorem of thelocal theory of curves 2E Curves in Minkowski space IR31 2F The global theory of curves Exercises Chapter 3. The Local Theory of Surfaces 3A Surface elements and the first fundamental form 3B The Gauss map and the curvature of surfaces 3C Surfaces of rotation and ruled surfaces 3D Minimal surfaces 3E Surfaces in Minkowski space IR31 3F Hypersurfaces in IRn+1 Exercises Chapter 4. The Intrinsic Geometry of Surfaces 4A The covariant derivative 4B Parallel displacement and geodesics 4C The Gauss equation and the Theorema Egregium 4D The fundamental theorem of the local theory of surfaces 4E The Gaussian curvature in special parameters 4F The Gauss-Bonnet Theorem 4G Selected topics in the global theory of surfaces Exercises Chapter 5. Riemannian Manifolds 5A The notion of a manifold 5B The tangent space 5C Riemannian metrics 5D The Riemannian connection Chapter 6. The Curvature Tensor 6A Tensors 6B The sectional curvature 6C The Ricci tensor and the Einstein tensor Chapter 7. Spaces of Constant Curvature 7A Hyperbolic space 7B Geodesics and Jacobi fields 7C The space form problem 7D Three-dimensional Euclidean and spherical space forms Exercises Chapter 8. Einstein Spaces 8A The variation of the Hilbert-Einstein functional 8B The Einstein field equations 8C Homogenous Einstein spaces 8D The decomposition of the curvature tensor 8E The Weyl tensor 8F Duality for four-manifolds and Petrov typesExercises
Solutions to selected exercises Bibliography List of notation Index
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