A metric is introduced on the space of parameters (couplings) describing the large N limit of the O(N) model in Euclidean space. The geometry associated with this metric is analysed in the particular case of the infinite volume limit in 3 dimensions and it is shown that the Ricci curvature diverges at the ultra-violet (Gaussian) fixed point but is finite and tends to constant negative curvature...

A Finsler space (M,Σ) is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason [13], it is shown that a geodesically reversible Finsler metric of constant flag curvature on the...

Let X be a geodesic space. We say that X is geodesically complete if every geodesic segment β : [0, a] → X from β(0) to β(a) can be extended to a geodesic ray α : [0,∞) → X, (i.e. β(t) = α(t), for 0 ≤ t ≤ a). If X is a compact npc space (“npc” means: “non-positively curved”) then it is almost geodesically complete, see [10]. (X, with metric d, is almost geodesically complete if its universal co...

The paper is devoted to metrization of probability spaces through the introduction of a quadratic differential metric in the parameter space of the probability distributions. For this purpose, a d-entropy functional is defined on the probability space and its Hessian along a direction of the tangent space of the parameter space is taken as the metric. The distance between two probability distri...

We use the Bradlow parameter expansion to construct the metric tensor in the space of solutions of the Bogomolny equations for the Abelian Higgs model on a twodimensional torus. Using this metric we study the dynamics and scattering of vortices on the torus within the geodesic approximation. For small torus volumes the metric is determined in terms of a small number of parameters. For large vol...

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the ...

In this paper, we present some common fixed point theorems for two generalized nonexpansive multivalued mappings in CAT(0) spaces as well as in UCED Banach spaces. Moreover, we prove the existence of fixed points for generalized nonexpansive multivalued mappings in complete geodesic metric spaces with convex metric for which the asymptotic center of a bounded sequence in a bounded closed convex...

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein’s model). Mathematics Subject Classi...

We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core. Let S be a surface of finite topological type, obtained by removing finitely many points from a compact surface without boun...

An n-dimensional polyhedral space is a length space M (with intrinsic metric) triangulated into n-simplexes with smooth Riemannian metrics. In the definitions below, we assume that the triangulation is fixed. The boundary of M is the union of the (n− 1)-simplexes of the triangulation that are adjacent to only one (n− 1)-simplex. As usual, a geodesic in M is a naturally parametrized locally shor...