An introduction to differential geometry with applications to mechanics and physics. It covers topology and differential calculus in banach spaces; differentiable manifold and mapping submanifolds; tangent vector space; tangent bundle, vector field on manifold, Lie algebra structure, and one-parameter group of diffeomorphisms; exterior differential forms; Lie derivative and Lie algebra; n-form integration on n-manifold; Riemann geometry; and more. It includes 133 solved exercises.
Table of Contents
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.
"The book is written in a very understandable and systematic way, with a lot of figures. A very good feature of the book is a collection of more than 130 exercises and problems … . The book can be recommended for a wide range of students as a first book to read on the subject. It can be also useful for the preparation of courses on the topic."
- EMS Newsletter
". . .a self-contained introduction to differential geometry. . ..very succinct."
---Monatshefte für Mathematik