On exotic spheres whose radial curvatures are close together

Kei Kondo, Minoru Tanaka

Published 2017-05-29Version 1

We prove that two compact manifolds are diffeomorphic if each of them admits a point whose cut locus consists of a single point, and if their radial curvatures are close together. Hence, our result produces a weak version of the Cartan--Ambrose--Hicks theorem in the case where underlying manifolds admit a point with a single cut point. In particular, that result generalizes one of theorems in Cheeger's Ph.D. Thesis in that case. Remark that every exotic sphere of dimension $> 4$ admits a metric such that there is a point whose cut locus consists of a single point.