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- Regina Rotman, R. ROTMAN
- 2008

In this paper we prove that given a point p ∈ M n , where M n is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop l p (M n) at this point is bounded above by 2nd, where d is the diameter of M n. Moreover, we show that on a closed simply connected Riemannian manifold M n with a non-trivial second homotopy group there either… (More)

- A Nabutovsky, R Rotman
- 2002

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemann-ian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M n is bounded from above by (n+2)!d 3 , where d is the diameter of a… (More)

- R. ROTMAN, Jon G. Wolfson
- 2006

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)

- Alexander Nabutovsky, Regina Rotman
- 2008

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)

- Alexander Nabutovsky, Regina Rotman
- 2007

Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M n. 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… (More)

- Alexander Nabutovsky, Regina Rotman, Albert Schwarz
- 2009

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)

- Alexander Nabutovsky, Regina Rotman
- 2009

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)

- Alexander Nabutovsky, Regina Rotman
- 2008

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 .

- REGINA ROTMAN
- 2007

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… (More)

- Alexander Nabutovsky, Regina Rotman
- 2008

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 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… (More)