Elementary differential geometry

Abstract

The model of spacetime used by the general theory of relativity is that of a differential manifold with a Riemannian geometry. In this and the next few chapters we develop the necessary background with emphasis on tools necessary for a physicist. This introduction is not rigorous from a mathematician’s point of view. We would assume that these manifolds have all the nice mathematical properties (Hausdorff nature, paracompactness etc.) which are needed for the existence of various geometrical quantities and procedures.