A Note on Closed Geodesics for a Class of non-compact Riemannian Manifolds

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits infinitely many prime closed geodesics such that the energy functional $E:\Lambda M\to\mathbb{R}$ has infinitely many non-degenerate critical points on the free loop space $\Lambda M$ of Sobolev class $H^1=W^{1,2}$. This result is obtained b...

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi Riemannian and magnetic structures on manifolds.

In this paper we prove a quantitative closing Lemma for manifolds of negative sectional curvature. As an application we study partner and pseudo-partner orbits for self-crossing closed geodesic.

We make some remarks on the existence of a geodesically complete core for any compact non-positively curved space.

The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe the...

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.

In this paper, we prove there are at least two closed geodesics on any compact bumpy Finsler $n$-manifold with $n\ge 2$. Thus generically there are at least two closed geodesics on compact Finsler manifolds. Furthermore, there are at least two closed geodesics on any compact Finsler $2$-manifold, and this lower bound is achieved by the Katok 2-sphere, cf. \cite{Kat}.

In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.

This paper sheds light on the essential characteristics of geodesics, which frequently occur in considerations from motion in Euclidean space. Focus is mainly on a method of obtaining them from the calculus of variations, and an explicit geodesic computation for a Riemannian hypersurface.

We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n>2.