Given two points of a Generalized Robertson-Walker spacetime, the existence, multiplicity and causal character of geodesics connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional spacetimes and to GRW spacetimes satisfying the timelike convergence conditi...

Given two points of a Generalized Robertson-Walker spacetime, the existence, multiplicity and causal character of geodesics connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional spacetimes and to GRW spacetimes satisfying the timelike convergence conditi...

It was shown by Chinburg and Reid that there exist closed hyperbolic 3-manifolds in which all closed geodesics are simple. Subsequently, Basmajian and Wolpert showed that almost all quasi-Fuchsian 3-manifolds have all closed geodesics simple and disjoint. The natural conjecture arose that the Chinburg-Reid examples also had disjoint geodesics. Here we show that this conjecture is both almost tr...

We study languages of geodesics in lamplighter groups and Thompson’s group F . We show that the lamplighter groups Ln have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group ...

We show that quasihyperbolic geodesics exist in convex domains in reflexive Banach spaces and that quasihyperbolic geodesics are quasiconvex in the norm metric in convex domains in all normed spaces. 2000 Mathematics Subject Classification: 30C65

We consider the scattering and lens rigidity of compact surfaces with boundary that have a trapped geodesic. In particular we show that the flat cylinder and the flat Möbius strip are determined by their lens data. We also see by example that the flat Möbius strip is not determined by it’s scattering data. We then consider the case of negatively curved cylinders with convex boundary and show th...

We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds P and Q of a manifold M endowed with a distribution D ⊂ TM. We give a different proof, that holds in a more general context, of a result by Bismut [2, Theorem 1.17] stating that the normal extremizers that are not abnormal are critical points of the sub-Riemannian action functio...

We propose a Fast Marching based implementation for computing sub-Riemanninan (SR) geodesics in the roto-translation group SE(2), with a metric depending on a cost induced by the image data. The key ingredient is a Riemannian approximation of the SR-metric. Then, a state of the art Fast Marching solver that is able to deal with extreme anisotropies is used to compute a SR-distance map as the so...

We associate certain probability measures on R to geodesics in the space H L of postively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H 0 (X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in H L as k goes to infinity. The convergence of secon...

Geodesics on the infinite dimensional symmetric space H of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in X×A, where A ⊂ C is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces GC/G. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Ampère geodesics c...