ON CLOSED GEODESICS IN THE LEAF SPACE OF SINGULAR RIEMANNIAN FOLIATIONS

In this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

Spectral flow and iteration of closed semi-Riemannian geodesics

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171-206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic gamma, we prove the existence of a locally constant integer valued map Lambda(gamma) on the unit circle with the property that the Morse index of the iterated gamma(N) is equal, up to a correction term epsilon(gamma) is an element of {0,1}, to the sum of the values of Lambda(gamma) at the N-th roots of unity. The discontinuities of Lambda(gamma) occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincare map of gamma. We discuss some applications of the theory.

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

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 nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics

Let (M, g) be a complete Riemannian Manifold, Omega subset of M an open subset whose closure is diffeomorphic to an annulus. If partial derivative Omega is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in (Omega) over bar = Omega boolean OR partial derivative Omega starting orthogonally to one connected component of partial derivative Omega and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a. class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

Geodesics and Almost Geodesic Cycles in Random Regular Graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d(G)(u, v) is at least d(C)(u, v) - e(n). Let omega(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= log(d-1)log(d-1) n+omega(n) and vertical bar C vertical bar =2 log(d-1) n+O(omega(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 66: 115-136, 2011

GENERICITY OF NONDEGENERATE GEODESICS WITH GENERAL BOUNDARY CONDITIONS

Let M be a possibly noncompact manifold. We prove, generically in the C(k)-topology (2 <= k <= infinity), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of L. Biliotti, M. A. Javaloyes and P. Piccione [6] for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P subset of M x M that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal Delta subset of M x M. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C(k)-generic are given.

de Sitter geodesics: reappraising the notion of motion

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Comment on "From geodesics of the multipole solutions to the perturbed Kepler problem"

In this Comment we explain the discrepancies mentioned by the authors between their results and ours about the in?uence of the gravitational quadrupole moment in the perturbative calculation of corrections to the precession of the periastron of quasielliptical Keplerian equatorial orbits around a point mass. The discrepancy appears to be a consequence of two different calculations of the angular momentum of the orbits.

Geodesics without differential equations: general relativistic calculations for introductory modern physics classes

Introductory courses covering modem physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.

Introduction to geodesics in sub-Riemannian geometry

International audience

Gravitational Waves: completeness and connectedness by geodesics

Only few months ago some physicists gave the official announcement that gravitational waves exist, but, from a geometrical point of view, they have always been ``real objects'' and their properties have been widely investigated. The aim of this talk is introducing generalized plane waves and discussing some of their properties such as geodesic connectedness and geodesic completeness.

A mathematical model of the early visual system and border completion via subriemannian Hamiltonian geodesics

In this thesis, we aim to discuss a simple mathematical model for the edge detection mechanism and the boundary completion problem in the human brain in a differential geometry framework. We describe the columnar structure of the primary visual cortex as the fiber bundle R2 × S1, the orientation bundle, and by introducing a first vector field on it, explain the edge detection process. Edges are detected through a lift from the domain in R2 into the manifold R2 × S1 and are horizontal to a completely non-integrable distribution. Therefore, we can construct a subriemannian structure on the manifold R2 × S1, through which we retrieve perceived smooth contours as subriemannian geodesics, solutions to Hamilton’s equations. To do so, in the first chapter, we illustrate the functioning of the most fundamental structures of the early visual system in the brain, from the retina to the primary visual cortex. We proceed with introducing the necessary concepts of differential and subriemannian geometry in chapters two and three. We finally implement our model in chapter four, where we conclude, comparing our results with the experimental findings of Heyes, Fields, and Hess on the existence of an association field.

Scattering of quantum particles by gravitational plane waves

We consider the coupling of quantum massless and massive scalar particles with exact gravitational plane waves. The cross section for scattering of the quantum particles by the waves is shown to coincide with the classical cross section for scattering of geodesics. The expectation value of the scalar field stress tensor between scattering states diverges at the points where classical test particles focus after colliding with the wave. This indicates that back-reaction effects cannot be ignored for plane waves propagating in the presence of quantum particles and that classical singularities are likely to develop.