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 Riemannian 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/4, where d is the diameter of a manifold M n . The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold M n is bounded from above by (n + 2)! FillRad(M n ) ≤ (n + 2)!(n + 1)n n (n + 1)!(vol(M n )) 1/n , where FillRad(M n ) is the filling radius of a manifold, and vol(M n ) is its volume.

Abstract. 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 Let l(M ) denote the length of a shortest closed non-trivial geodesic on a closed Riemannian manifold M and let A be the area of M . In this paper we will prove the following theorem.Theorem 0

In this paper we will show that for any 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 them of length at most 22kd, where d is the diameter of M .Introduction.

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 unless the length functional has "many" "deep" local minima of a "small" length on the space Ω pq M n of paths connecting p and q. For example, one of our results implies that for every positive integer k there are two possibilities: Either the length functional on Ω pq M n has k distinct non-trivial local minima with length ≤ 2kd and "depth" ≥ 2d; or for every m every map of S m into Ω pq M n is homotopic to a map of S m into the subspace Ω 4(k+2)(m+1)d pq M n of Ω pq M n that consists of all paths of length ≤ 4(k+ 2)(m + 1)d.