The convexity radius of a Riemannian manifold

Abstract

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characterization of the convexity radius that resembles a classical result of Klingenberg about the injectivity radius.

arXiv comment6 pages; corrected false statements in the first and third sections and an error in Lemma 2.4; removed one reference and added three others; made other minor stylistic edits