Manifolds without 1/k-geodesic

Abstract

It is a question by C.Sormani that whether there exists a $k\in \mathbb{N}$, such that any compact, smooth and simply connected manifold has a 1/k-geodesic. We prove in this paper that this is not true by showing for each $k$, there exists a metric on the sphere such that it has no 1/k-geodesic.

11 pages, 8 figures