Regina Rotman

The length of a shortest closed geodesic and the area of a 2-dimensional sphere

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.B. Croke (1988), by A. Nabutovsky and the author (2002) and by S. Sabourau (2004).

The length of the second shortest geodesic

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 diameterd of the manifold. But how long are the shortest remaining geodesics? In this paper we prove that any two points on a closed n-dimensional Riemannian manifold can be connected by two distinct geodesics of length � 2qd � 2nd, where q is the smallest value of i such that the i th homotopy group of the manifold is non- trivial.

Lengths of geodesics between two points on a Riemannian manifold

Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold Mn. According to a well-known theorem by J.-P. Serre, there exist infinitely many geodesics between x and y. It is obvious that the length of a shortest of these geodesics cannot exceed the diameter of the manifold. But what can be said about the lengths of the other geodesics? We conjecture that for every k there are k distinct geodesics of length ≤ k diam(Mn). This conjecture is evidently true for round spheres and is not difficult to prove for all closed Riemannian manifolds with non-trivial torsionfree fundamental groups. In this paper we announce two further results in the direction of this conjecture. Our first result is that there always exists a second geodesic between x and y of length not exceeding 2n diam(Mn). Our second result is that if n = 2 and M2 is diffeomorphic to S2, then for every k every pair of points of M2 can be connected by k distinct geodesics of length less than or equal to (4k2 − 2k − 1)diam(M2).

LINEAR BOUNDS FOR LENGTHS OF GEODESIC SEGMENTS ON RIEMANNIAN 2-SPHERES

In this paper we will show that for every positive integer k and each pair of points p, q ∈ M, where M is a Riemannian manifold diffeomorphic to the 2-dimensional sphere, there always exist at least k geodesics connecting p and q of length at most 22kd, where d is the diameter of M.

LENGTHS OF SIMPLE PERIODIC GEODESICS ON TWO-DIMENSIONAL RIEMANNIAN SPHERES

Let M be a Riemannian two-sphere. A classical result of Lyusternik–Shnirelman asserts the existence of three distinct simple nontrivial periodic geodesics on M. In this paper, we will prove that the lengths of two of them do not exceed 5d and 10d respectively, where d is the diameter of M.

Short geodesic loops on complete Riemannian manifolds with a finite volume

In this paper we will show that on any complete noncompact Riemannian manifold with a finite volume there exist uncountably many geodesic loops of arbitrarily small length.