Francisco J. Palomo

A Berger-Green type inequality for compact Lorentzian manifolds

We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.

Lorentzian manifolds with no null conjugate points

An integral inequality for a compact Lorentzian manifold which admits a timelike conformal vector field and has no conjugate points along its null geodesics is given. Moreover, equality holds if and only if the manifold has nonpositive constant sectional curvature. The inequality can be improved if the timelike vector field is assumed to be Killing and, in this case, the equality characterizes (up to a finite covering) flat Lorentzian n( 3)-dimensional tori. As an indirect application of our technique, it is proved that a Lorentzian 2−torus with no conjugate points along its timelike geodesics and admitting a timelike Killing vector field must be flat.

On spacelike surfaces in four-dimensional Lorentz–Minkowski spacetime through a light cone

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann type result. Two remarkable families of examples of spacelike surfaces in a lightcone are explicitly constructed. Finally, several results which involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a lightcone are obtained.

Conformally stationary Lorentzian tori with no conjugate points are flat

A Lorentzian torus which admits a timelike conformal vector field and with no conjugate points on its timelike and spacelike geodesics is proved to be flat. If only the absence of conjugate points on timelike geodesics is assumed, a counterexample is shown.

Chapter 8 Certain actual topics on modern Lorentzian geometry

Publisher Summary This chapter discusses the research on four relevant topics on Lorentzian geometry. Several Lorentzian results are compared to Riemannian or indefinite (non-Lorentzian) ones, emphasizing on mathematical behaviors, which are specific of Lorentzian geometry. Degenerate tangent planes play an important role in the study of the geometry of Lorentzian manifolds. Harris introduced the notion of null sectional curvature for degenerate tangent planes, which has shown to be fruitful to get some comparison theorems and to characterize Robertson–Walker spacetimes. The chapter discusses Bochner technique on Lorentzian manifolds focusing aims and difficulties.

Erratum to ``A Berger-Green type inequality for compact Lorentzian manifolds"

Proof of Proposition 2.3. Let V ∈ X(TM) be given by ĝ(Vv, X) = g(c(X),Kπv) + g(v,∇dπ(X)K), for every X ∈ TvTM . Thus we have ĝ(Vv,Av) = 1, at any v ∈ CK(M), and so Dv = Span{Vv,Av} is a Lorentzian vector subspace of TvTM (recall that ĝ(A,A) = 0 on CKM). One easily checks that X ∈ TvCKM if and only if ĝ(Av, X) = ĝ(Vv, X) = 0. Therefore, (CKM, ĝ) is a Lorentzian manifold. The proof for spacelike fibres remains valid. Finally, [Tv(CKM)p] = {X ∈ TvTM : c(X) = −g(v,∇dπ(X)K) v}, and so π restricted to CKM is a semi-Riemannian submersion.

A METHOD TO CONSTRUCT 4-DIMENSIONAL SPACETIMES WITH A SPACELIKE CIRCLE ACTION

A general procedure to construct a 4-dimensional spacetime from a 3-dimensional time-oriented Lorentzian manifold and each of its timelike vector fields is exposed. It is based on the construction of the null congruence Lorentzian manifold. As an application, examples of stably causal spacetimes which obey the timelike convergence condition, are semi-symmetric, and admit an isometric spacelike circle action are obtained.