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Length of geodesics on a two-dimensional sphere


Let $M$ be an arbitrary Riemannian manifold diffeomorphic to $S^2$. Let $x,y$ be two arbitrary points of $M$. We prove that for every $k=1,2,3,\ldots$ there exist $k$ distinct geodesics between $x$ and $y$ of length less than or equal to $(4k^2-2k-1)d$, where $d$ denotes the diameter of $M$.

To prove this result we demonstrate that for every Riemannian metric on $S^2$ there are two (not mutually exclusive) possibilities: either every two points can be connected by many ``short" geodesics of index $0$, or the resulting Riemannian sphere can be swept-out by ``short meridians".