Differential Geometry with Applications to Mechanics and Physics

Abstract

Part 1 Topology and differential calculus requirements: topology differential calculus in Banach spaces exercises. Part 2 Manifolds: introduction differential manifolds differential mappings submanifolds exercises. Part 3 Tangent vector space: tangent vector tangent space differential at a point exercises. Part 4 Tangent bundle-vector field-one-parameter group lie algebra: introduction tangent bundle vector field on manifold lie algebra structure one-parameter group of diffeomorphisms exercises. Part 5 Cotangent bundle-vector bundle of tensors: cotangent bundle and covector field tensor algebra exercises. Part 6 Exterior differential forms: exterior form at a point differential forms on a manifold pull-back of a differential form exterior differentiation orientable manifolds exercises. Part 7 Lie derivative-lie group: lie derivative inner product and lie derivative Frobenius theorem exterior differential systems invariance of tensor fields lie group and algebra exercises. Part 8 Integration of forms: n-form integration on n-manifold integral over a chain Stokes' theorem an introduction to cohomology theory integral invariants exercises. Part 9 Riemann geometry: Riemannian manifolds.