We recall in this chapter basic facts concerning Riemannian geometry and nonlinear analysis on manifolds. For reasons of length, we are obliged to be succinct and partial. Possible references are Chavel , do Carmo , Gallot-Hulin-Lafontaine , Hebey , Jost , Kobayashi-Nomizu , Sakai , and Spivak .As a general remark, we mention that Einstein’s summation convention is adopted:an index occurring twice in a product is to be summed. This also holds for the rest of this book.
1.1 RIEMANNIAN GEOMETRY
We refer to the above definition of a manifold as the abstract definition of a smooth manifold. ...