Curvature in Mathematics and Physics by Shlomo Sternberg

Author:Shlomo Sternberg
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2013-06-26T16:00:00+00:00

where k is the number of subintervals into which we have divided [0, 1],

For large enough j we have l(Cj) − d < and

d(Cj(t),C(t)) < ∀t ∈ [0,1].

Let denote the restriction of Cj to the interval [ti−1, ti]. Then

Therefore there is some i such that For large enough j we get a path joining the end points of γi of shorter length, contradicting the choice of γi.

Chapter 9

The Hopf-Rinow theorem.

This chapter is devoted to a theorem of Hopf and Rinow which relates the various concepts of “completeness” in Riemannian geometry. In this chapter M will be a connected Riemannian manifold (with the exception of the beginning of §9.4 where semi-Riemannian manifolds are allowed).

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