A geodesic is a parameterized curve on a Riemannian manifold governed by a second order partial differential equation. Geodesics are notoriously unstable: small perturbations of the underlying manifold may lead to dramatic changes of the course of a geodesic. Such instability makes it difficult to use geodesics in many applications, in particular in the world of discrete geometry. In this paper...

Let X be a compact Riemannian manifold with conic singularities, i.e. a Riemannian manifold whose metric has a conic degeneracy at the boundary. Let ∆ be the Friedrichs extension of the Laplace-Beltrami operator on X. There are two natural ways to define geodesics passing through the boundary: as “diffractive” geodesics which may emanate from ∂X in any direction, or as “geometric” geodesics whi...

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 ...

The condition metric in spaces of polynomial systems has been introduced and studied in a series of papers by Beltrán, Dedieu, Malajovich and Shub. The interest of this metric comes from the fact that the associated geodesics avoid ill-conditioned problems and are a useful tool to improve classical complexity bounds for Bézout’s theorem. The linear case is examined here: Using nonsmooth nonconv...

This paper is devoted to a study of geodesics of Finsler metrics via Zermelo navigation. We give a geometric description of the geodesics of the Finsler metric produced from any Finsler metric and any homothetic field in terms of navigation representation, generalizing a result previously only known in the case of Randers metrics with constant S-curvature. As its application, we present explici...

In this paper we prove the existence of closed geodesics in certain types of compact Riemannian good orbifolds. This gives us an elementary alternative proof of a result due to Guruprasad and Haefliger. In addition, we prove some results about horizontal periodic geodesics of Riemannian foliations and stress the relation between them and closed geodesics in Riemannian orbifolds. In particular w...

It is an important problem to compute the geodesics on a surface in many fields. To find the geodesics in practice, however, the traditional discrete algorithms or numerical approaches can only find a list of discrete points. The first author proposed in 2010 a new, elegant and accurate method, the geodesic-like method, for approximating geodesics on a regular surface. This paper will present b...

There is a general method, applicable in many situations, whereby a pseudo–Riemannian metric, invariant under the action of some Lie group, can be deformed to obtain a new metric whose geodesics can be expressed in terms of the geodesics of the old metric and the action of the Lie group. This method applied to Euclidean space and the unit sphere produces new examples of complete Riemannian metr...