In this paper we prove some results on the number of geodesics connecting two points or two submanifolds on a non-reversible complete Finsler manifold, in particular for complete Randers metrics. We apply the abstract results to the study of light rays and timelike geodesics with fixed energy on a standard stationary Lorentzian manifold.

Three Dimensional Manifolds All of Whose Geodesics Are Closed John Olsen Wolfgang Ziller, Advisor We present some results concerning the Morse Theory of the energy function on the free loop space of S for metrics all of whose geodesics are closed. We also show how these results may be regarded as partial results on the Berger Conjecture in dimension three.

Given a C∞ Riemannian metric g on RP 2 we prove that (RP , g) has constant curvature iff all geodesics are closed. Therefore RP 2 is the first non trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large set of C∞ metric...

Let M be a compact simply connected manifold satisfying H⁎(M;Q)≅Td,n+1(x) for integers d≥2 and n≥1. If all prime closed geodesics on (M,F) with an irreversible bumpy Finsler metric F are elliptic, then either there exist exactly dn(n+1)2 (when is even) or (d+1) d≥3 odd) distinct geodesics, infinitely many geodesics.

In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Tod and ...

For all systolic groups we construct boundaries which are EZ–structures. This implies the Novikov conjecture for torsion–free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT (0) spaces. MSC: 20F65; 20F67; 20F69;

Existence and uniqueness of complex geodesics joining two points of a convex bounded domain in a Banach space X are considered. Existence is proved for the unit ball of X under the assumption that X is 1-complemented in its double dual. Another existence result for taut domains is also proved. Uniqueness is proved for strictly convex bounded domains in spaces with the analytic Radon-Nikodym pro...

In this paper, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, Möbius band plane leads to infinitely many geodesics. particular, prove that any cylinder with isolated has zero, homologically visible geodesics; answers question of Alberto Abbondandolo.

We consider a wide class of ergodic first passage percolation processes on Z2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four color Richardson’s growth model. This improves earlier results of Häggström and Pemantle [9], Garet and Marchand [7] and Hoffman [11] who proved that first passage percolation...

In this paper we obtain asymptotic estimates for pairs of closed geodesics on negatively curved manifolds, the differences of whose lengths lie in a prescribed family of shrinking intervals, were the geodesics are ordered with respect to a discrete length. In certain cases, this discrete length can be taken to be the word length with respect to a set of generators for the fundamental group.