This book arose out of courses taught by the author. It covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The author emphasizes geometric concepts, giving the reader a working knowledge of the topic. Motivations are given, exercises are included, and illuminating nontrivial examples are discussed.
Important features include the following:
Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples.
A thorough discussion of the much-used result on the existence, uniqueness, and smooth dependence of solutions of ODEs.
Careful introduction of the concept of tangent spaces to a manifold.
Early and simultaneous treatment of Lie groups and related concepts.
A motivated and highly geometric proof of the Frobenius theorem.
A constant reconciliation with the classical treatment and the modern approach.
Simple proofs of the hairy-ball theorem and Brouwer's fixed point theorem.
Construction of manifolds of constant curvature a la Chern.
This text would be suitable for use as a graduate-level introduction to basic differential and Riemannian geometry.
