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Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
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Geodesics of bounded length on a Riemannian manifold
Does there exist a compact Riemannan manifold $M^n$ and an $L > 0$ such that the number of homotopy classes of simple closed curves $\gamma$ on $M^n$ whose shortest representatives have length at most …