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

The results of this paper have been greatly superseded by those in the paper "Contact geometry and isosystolic inequalities" (arXiv:1109.4253) by the same authors.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

This article concerns a class of metric spaces, which we call multigeodesic spaces, where between any two distinct points there exist multiple distinct minimising geodesics. We provide a simple characterisation of multigeodesic normed spaces and deduce that $(C([0,1]),||\cdot||_1)$ is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.

In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply ...

We construct a compact semi-Riemannian manifold without periodic geodesics. We also provide examples of three-dimensional Lorentz manifolds without closed geodesics. All these examples are of the form $\Gamma\backslash G$ where $G$ is a Lie group endowed with a left invariant metric and $\Gamma\subset G$ is a cocompact lattice. Unlike the locally homogeneous case, we show that a homogeneous semi-Riemannian manifold admits closed geodesics.

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics

We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.

The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities.