2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/117987It 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 figuresDifferential Geometry53C20text