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.... 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".