Lectures on Differential Geometry

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.