A Course in Differential Geometry

Explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. The textbook deals with vector fields and differential forms, and also addresses integration of vector fields and p-planes.

This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. It explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. It deals with vector fields and differential forms, addresses integration of vector fields and p-planes and develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative, as well as specializing on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely and explores some problems in PDEs suggested by the geometry of manifolds.