Asian Journal of Mathematics

Volume 21 (2017)

Number 1

The convexity radius of a Riemannian manifold

Pages: 169 – 174



James Dibble (Department of Mathematics, Rutgers University–New Brunswick, Piscataway, New Jersey, U.S.A.; and Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)


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.


convexity radius, injectivity radius, focal points, conjugate points

2010 Mathematics Subject Classification


Published 16 March 2017