Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

Connections on Naturally Reductive

Simple Closed Geodesics in Hyperbolic 3-Manifolds

This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete nonelementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite or that it has finitely generated fundament...

Non-simple Geodesics in Hyperbolic 3-manifolds

Chinburg and Reid have recently constructed examples of hyperbolic 3manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain an immersed totally geodesic surf...

Drilling long geodesics in hyperbolic 3-manifolds

Given a complete hyperbolic 3-manifold one often wants to compare the original metric to a complete hyperbolic metric on the complement of some simple closed geodesic in the manifold. In some cases this can be done by interpolating between the two metrics using hyperbolic cone-manifolds. We refer to such a deformation as drilling and results which compare the geometry of the original manifold t...

Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.