We study geodesics on planar Riemann surfaces of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing the surface.

We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set of all points that are connected to the root by more than one geodesic. We also prove that points of the Brownian map can be connected to ...

The scattering data of a Riemannian manifold with boundary record the incoming and outgoing directions of each geodesic passing through. We show that the scattering data of a generic Riemannian surface with no trapped geodesics and no conjugate points determine the lengths of geodesics. Counterexamples exists when trapped geodesics are allowed.

We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank. The total space is the general linear group endowed with its natural rightinvariant metric, and the metric on the homogeneous space is chosen such that the quotient space is the image of a Riemannian submersion from the total space. As a result, we obtain complete geodesics that a...

Let S be a surface and let P be a pair of pants. Geodesics on surfaces, and on pairs of pants specifically, have been studied extensively over the years. In this paper, we focus on getting a lower bound on the number of closed geodesics on P with given upper bounds on length and self-intersection number. As a direct consequence, we get a lower bound for the number of such closed geodesics on an...

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...

*Correspondence: Valter Moretti, Dipartimento di Matematica, Università di Trento and Istituto Nazionale di Fisica Nucleare Gruppo Collegato di Trento, via Sommarive 14 I-38050 Povo, Italy e-mail: [email protected] A spacetime is locally flat if and only if no geodesical deviation exists for congruences of all kinds of geodesics. However, while for causal geodesics the deviation can be m...

We examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M . In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using techniques from analytic number theory, we address the following problems: Is the commensurability class of...

In Euclidean space, the geodesics on a surface of revolution can be characterized by means of Clairaut’s theorem, which essentially says that the geodesics are curves of fixed angular momentum. A similar result is known for three dimensional Minkowski space for timelike geodesics on surfaces of revolution about the time axis. Here, we extend this result to consider generalizations of surfaces o...

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid Q that are cut out by algebraic surfaces in R. Such geodesics are either connected components of spatial elliptic curves or of rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves, the addition...