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Length of geodesics on a two-dimensional sphere

Volume 131, Number 2, April 2009
pp. 545-569 | 10.1353/ajm.0.0042

Abstract

Abstract:

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".

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