Manifolds without \(\tfrac {1}{k}\)-geodesics
Let M be a smooth hypersurface in the Euclidean space and \(\gamma: S^1\rightarrow M\) a closed geodesic parametrized by arc length and with length \(l\). Then \(\gamma\) is called a \(1/k\)-geodesic if it is distance minimizing on every segment of length \(l/k\). In the paper it is shown that for every integer \(k\geq 2\) there exists a smooth surface (a surface of revolution) \(M_k\) which has no \(1/k\)-geodesic.
scientific article; zbMATH DE number 5508722