Lectures on Differential Geometry.

Abstract

Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.