Geodesics in semi-Riemannian manifolds: geometric properties and variational tools

Abstract

Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially relevant the systematic introduction of (infinite-dimensional) variational methods.Our purpose is to give an overview, from refinements of classical results to updated variational settings. First, several properties (and especially completeness) of geodesics in some ambient spaces are studied. This includes heuristic constructions of compact incomplete examples, geodesics in warped, GRW or stationary spacetimes, properties in surfaces and spaceforms, or problems on stability of completeness.Then, we study the variational framework, and focus on two fundamental problems of this approach, which regards geodesic connectedness. The first one deals with a variational principle for stationary manifolds, and its recent implementation inside Causality Theory. The second one concerns orthogonal splitting manifolds, and a reasonably selfcontained development is provided, collecting some steps spread in the literature.

An Invitation to Lorentzian Geometry

“…In this section, we briefly recall the basics of Lorentzian geometry and compare them to the Riemannian situation (see [17,28,48,83,89,114] for further details). Let, throughout this article, M be an n-dimensional manifold, oriented if necessary.…”

“…in spaceforms and other physical or mathematically relevant spacetimes; notice that some of these questions had motivations from the viewpoint of the initial value problem and were commented in Section 5, but such problems evolve further, independent of physical motivations. (2) Critical curves for indefinite functionals on Lorentzian manifolds: even though the role of geodesics in General Relativity gives a general support for this, the infinite-dimensional variational mathematical approach for geodesics, including spacelike ones, has independent interest, see the seminal works by Benci, Fortunato and Giannoni [19], the book [81] of the review [28]; we emphasize that even a simple question as if any compact Lorentzian manifold must admit a closed geodesic remains open. (3) Curvature: curvature bounds groups have been stressed above, but there are many other questions related with curvature operators, e.g., those starting at the Osserman problem, solved a decade ago, see [58].…”

“…As shown in [29], a very precise result on the existence and multiplicity of light rays from c to p in physically realistic spacetimes, can be stated in terms of the geodesics connecting two points for an appropriate Finsler metric. For the variational study of spacelike geodesics, see [81,28] and references therein.…”

“…So, Causality can be identified to conformal geometry in Lorentzian signature (nevertheless, a subtler modification of the notion of Causality has been recently introduced by García-Parrado and Senovilla [57], see also [56]). Remarkably, one can associate a (conformally invariant) causal boundary to every sufficiently well behaved spacetime, this allows to describe the possible asymptotics of timelike curves in a subtle way, ordering them by inclusion of their pasts [50] As a remarkable difference with the Riemannian case, both, completeness and incompleteness are C r unstable, for every r ∈ N, even in the case of Lorentzian metrics on a torus [99], but some results on stability can be still obtained in the Lorentzian case, as C 1 -fine stability in the globally hyperbolic case, see [17,28]. The interplay between these particularities, causality and some physical interpretations, have the effect that, in Lorentzian Geometry, singularity theorems (which ensure incompleteness rather than completeness) play an important role, see below.…”

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

“…Indeed Lorentzian warped products of geodesically complete manifolds need not be complete, as occurs in positive definite signature. Necessary and sufficient conditions for geodesic completeness of Lorentzian warped products were investigated in [9]. Let M = I × ω N be a warped product where I = (α, β) is a real interval and (N, g N ) is a geodesically complete manifold.…”