Manifolds without 1/k-geodesic
Abstract
It is a question by C.Sormani that whether there exists a k ∈ 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.
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