Chapter One

Background Material

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 [20], do Carmo [22], Gallot-Hulin-Lafontaine [36], Hebey [43], Jost [50], Kobayashi-Nomizu [53], Sakai [65], and Spivak [72].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

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We refer to the above definition of a manifold as the abstract definition of a smooth manifold. ...

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