The convexity radius of a Riemannian manifold

TL;DR

This paper demonstrates that in compact Riemannian manifolds of dimension two or higher, the ratio of convexity radius to injectivity radius can be arbitrarily small, using manifolds with focal points but no conjugate points.

Contribution

It introduces a method to construct manifolds with focal points but no conjugate points, showing the ratio of convexity radius to injectivity radius can be arbitrarily small.

Findings

The ratio of convexity radius to injectivity radius can be made arbitrarily small.

Construction of manifolds with focal points but no conjugate points.

A characterization of the convexity radius similar to Klingenberg's result.

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.