Regina Rotman

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According to the classical result of J.P. Serre ([S]) any two points on a closed Riemannian manifold can be connected by infinitely many geodesics. The length of a shortest of them trivially does not exceed the diameter of the manifold, d. But how long are the shortest remaining geodesics? In this paper we prove that any two points on a closed n-dimensional(More)
Let M be a Riemannian manifold homeomorphic to S 2. The purpose of this paper is to establish the new inequality for the length of a shortest closed geodesic, l(M), in terms of the area A of M. This result improves previously known inequalities by C. Let l(M) denote the length of a shortest closed non-trivial geodesic on a closed Riemannian manifold M and(More)
Let M n be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points p, q ∈ M n and every positive integer k there are at least k distinct geodesics connecting p and q of length ≤ 4nk 2 d. We demonstrate that all homotopy classes of M n can be represented by spheres swept-out by " short " loops(More)
Lengths of geodesics between two points on a Riemannian manifold. Abstract. Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M n. According to a celebrated theorem by J.P. Serre there exist infinitely many geodesics between x and y. The length of the shortest of these geodesics is obviously less than the diameter of the(More)
Let M n be a closed Riemannian manifold homotopy equivalent to the product of S 2 and an arbitrary (n − 2)-dimensional manifold. In this paper we prove that given an arbitrary pair of points on M n there exist at least k distinct geodesics of length at most 20k!d between these points for every positive integer k. Here d denotes the diameter of M n .
Let D be a Riemannian 2-disc of area A, diameter d and length of the boundary L. We prove that it is possible to contract the boundary of D through curves of length ≤ L+200d max{1, ln √ A d }. This answers a twenty-year old question of S.Frankel and M.Katz, a version of which was asked earlier by M.Gromov. We also prove that a Riemannian 2-sphere M of(More)