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...

Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with labeled angles and edges. In the case of the Coxeter geodesic domes, the plane graph is the icosahedron, all ang...

For a space–time, the question whether two causally related points can be joined by means of a causal geodesic has a clear physical meaning. More geometrically, geodesic connectedness ~the possibility of joining any two points by a geodesic of any causal type! is a basic property. Some techniques introduced to study geodesic connectedness are appliable to related problems of physical interest, ...

A general class of Lorentzian metrics, M0 × R 2 , ·, ·z = ·, ·x + 2 du dv + H(x, u) du 2 , with (M0, ·, ·x) any Riemannian mani-fold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity or causal character of connecting geodesics. These result...

This article contains a detailed study in the case of a toric variety of the geodesic rays φt defined by Phong-Sturm corresponding to test configurations T in the sense of Donaldson. We show that the ‘Bergman approximations’ φk(t, z) of Phong-Sturm converge in C to the geodesic ray φt, and that the geodesic ray itself is C 1,1 and no better. In particular, the Kähler metrics ωt = ω0 + i∂∂̄φt ass...

The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler–Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given, especially in the special case of easy totally geodesic submanifolds that we introd...

In this paper we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let Mn be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that ...

This paper describes the Subgradient Marching algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N log(N)) operations on a discrete grid ofN points. It performs a front propagation that computes the subgradient of a discrete geodesic dist...

Let P be a simple polygon of n vertices and let S be a set of n points called sites lying in the interior of P. The geodesic-distance between two sites x and y in P is defined as the length of the shortest polygonal path connecting x and y constrained to lie in P. It is a useful notion in graphics problems concerning visibility between objects, computer vision problems concerned with the descri...