Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold X . We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesi...

Communicated May 10, 1960 1. This paper deals with a space V4 of four dimensions which admits minimal geodesics as paths of rays of lights, that is curves for which dx? dxi 9ij d ° 1 ds s(1 This means that the symmetric metric tensor gij is not positive definite. The equation of geodesics in V4 iSl d2Xi dxi dXk ds2 + jkf ds ds (2) The second term in this equation stands for the sum of terms as ...

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as ...

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X, dN ) a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics. We introduce two assumptions on the transference plan π whic...

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to...

In this work an adaptive strategy for the phase space method [5] for traveltime tomography is developed. The method uses those geodesics/rays that produces smaller mismatch with the measurements first and continue on in the spirit of layer stripping without the need to specifically defining the layers physically. The adaptive approach improves stability, efficiency and accuracy. We then extend ...

In this paper, we have studied QNM modes and absorption cross sections of Born-Infeld-de Sitter black holes. WKB approximation is employed to compute the QNM modes of massless scalar fields. We have also used null geodesics to compute quasi-normal modes in the eikonal approximation. In the eikonal limit QNMs of black holes are determined by the parameters of the circular null geodesics. Unstabl...

Homogeneous sub-Riemannian geodesics are described for the standard structure on group $${\mathrm {SE}}(2)$$ of proper motions plane. It is shown that this not geodesically orbital, despite invariance cut time when initial point shifted along geodesics.