CONVEXITY STUDY IN RIEMANNIAN MANIFOLDS

Abstract

This chapter reviews briefly the standard concepts and theorems of differential geometry that will be needed in he main part of this work.

1.1 Manifolds and Submanifolds [17].

••.Let r be an integer, r > 0. A map f from an open set A c Rn into R is called cr on A if it possesses continuous partial derivatives on A of all orders :s r; If f is continuous from A to R, .then f is co on tl1 • . If f is cr on A for all r, then f is c"' on A

Let M be a set. An n-coordinate pair on M is a pair

1-, U) consisting of a subset U of M and a one-one map ¢ of U

onto an open set in Rn. One n-coordinate pair (¢, U) is Cr

related to another n-coordinate pair (0, VI, such that UrN *¢

. - 1 if the maps ¢ o 0 and 0 o

rp - 1

are Cr

maps .(See Fig. 1.1).

----..._ M -

-------.-•- ---- --F-i-g-.--(-1-.-1-) ---•

A cr n-subatlas on M is a collection of n coordinate pairs