An Introduction to Riemannian Geometry
Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity.
The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects.
The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.
Presents a self-contained treatment of Riemannian geometry and applications to mechanics and relativity in one bookConveys nontrivial results in general relativity (such as the Hawking and Penrose singularity theorems) which are not usually treated in introductory textsContains detailed solutions to many of the 300 exercises to help students test and consolidate their understandingIncludes a summary of all the main definitions and results from the necessary background material in differential calculus, algebra and topology