Local and global convexity in complete Riemannian manifolds

Stephanie Brewster Brewer Taylor Alexander

Vol. 76 (1978), No. 2, 283–289

DOI: 10.2140/pjm.1978.76.283

Abstract

A connected open set in Euclidean space is convex if it is locally supported at each boundary point; indeed, the same statement holds in any complete Riemannian manifold for which all geodesics are minimal. On the other hand, in an arbitrary complete n-dimensional Riemannian manifold M the question, under what circumstances global convexity properties are implied by local ones, involves the notion of cut locus. This question will be considered here.

Mathematical Subject Classification 2000

Primary: 53C45

Milestones

Received: 28 December 1976

Published: 1 June 1978