Geometry

Abstract

We collect here some basic facts and principles of differential geometry as the foundation for the sequel. For a more penetrating discussion and for the proofs of various results, we refer to J. Jost (Riemannian Geometry and Geometric Analysis, 5th edn., Springer, Berlin, 2008). Classical differential geometry as expressed through the tensor calculus is about coordinate representations of geometric objects and the transformations of those representations under coordinate changes. The geometric objects are invariantly defined, but their coordinate representations are not, and resolving this contradiction is the content of the tensor calculus.