Length of geodesics on a two-dimensional sphere

  title={Length of geodesics on a two-dimensional sphere},
  author={Alexander Nabutovsky and Regina Rotman},
Let M be an arbitrary Riemannian manifold diffeomorphic to S. Let x, y be two arbitrary points of M . We prove that for every k = 1, 2, 3, . . . there exist k distinct geodesics between x and y of length less than or equal to (4k − 2k− 1)d, where d denotes the diameter of M . To prove this result we demonstrate that for every Riemannian metric on S there are two (not mutually exclusive) possibilities: either every two points can be connected by many “short” geodesics of index 0, or the… CONTINUE READING