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 book Conveys nontrivial results in general relativity (such as the Hawking and Penrose singularity theorems) which are not usually treated in introductory texts Contains detailed solutions to many of the 300 exercises to help students test and consolidate their understanding Includes a summary of all the main definitions and results from the necessary background material in differential calculus, algebra and topology Includes supplementary material: sn.pub/extras Request lecturer material: sn.pub/lecturer-material

Autorentext

Leonor Godinho is professor at Instituto Superior Técnico (Universidade de Lisboa). She regularly teaches Riemannian geometry, symplectic geometry and introductory geometry courses. Her research activity is focused on symplectic geometry and its connections to algebraic geometry and combinatorics.

José Natário is professor of mathematics at Instituto Superior Técnico (Universidade de Lisboa). He regularly lectures on differential and Riemannian geometry, geometric mechanics and mathematical relativity. His research focuses on general relativity, a subject on which he has published many research papers and a book, General Relativity Without Calculus (Springer, 2011).

Inhalt
Differentiable Manifolds.- Differential Forms.- Riemannian Manifolds.- Curvature.- Geometric Mechanics.- Relativity.