Lorentzian Geometry of the Heisenberg Group

In this work we prove the existence of totally geodesic two-dimensional foliation on the Lorentzian Heisenberg group H ₃. We determine the Killing vector fields and the Lorentzian geodesics on H ₃.     

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 γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

Induced Riemannian structures on null hypersurfaces

Given a null hypersurface L of a Lorentzian manifold, we construct a Riemannian metric on it from a fixed transverse vector field ζ. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold and the vector field ζ. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.     

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.     

Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime

We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so‐called generalized Robertson‐Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett  29 .     

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02     

Global decomposition of a Lorentzian manifold as a Generalized Robertson-Walker space

Generalized Robertson-Walker (GRW) spaces constitute a quite important family in Lorentzian geometry, and it is an interesting question to know whether a Lorentzian manifold can be decomposed in such a way. It is well known that the existence of a suitable vector field guaranties the local decomposition of the manifold. In this paper, we give conditions on the curvature which ensure a global decomposition and apply them to several situations where local decomposition appears naturally. We also study the uniqueness question, obtaining that the de Sitter spaces are the only nontrivial complete Lorentzian manifolds with more than one GRW decomposition. Moreover, we show that the Friedmann Cosmological Models admit an unique GRW decomposition, even locally.     

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Lightlike surfaces of spacelike curves in de Sitter 3-space

The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces. Work partially supported by Grant-in-Aid for formation of COE. ‘Mathematics of Nonlinear Structure via Singularities’     

Uniqueness of Lorentzian Hopf tori

We prove that Lorentzian Hopf tori are the only immersed Lorentzian flat tori in a wide family of Lorentzian three-dimensional Killing submersions with periodic timelike orbits.     

Homogeneous Geodesics of Three-Dimensional Unimodular Lorentzian Lie Groups

We consider three-dimensional unimodular Lie groups equipped with a Lorentzian metric and we determine, for all of them, their sets of homogeneous geodesics through a point. Dedicated to the memory of Professor Aldo Cossu Authors supported by funds of M.U.R.S.T., G.N.S.A.G.A. and the University of Lecce.     

Maslov index and Morse theory for the relativistic Lorentz force equation

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.Mathematics Subject Classification (2002): Primary: 58E10, 83C10; Secondary: 53D12     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

s-Representations for involutions¶on affine Kac–Moody algebras are polar

Let denote the eigenspace decomposition of a twisted affine Kac–Moody algebra with respect to an involution , where is a twisted loop algebra, is the center and d is the scaling element of . We endow with the standard bilinear symmetrical form.Then with and carries a Lorentzian signature. Let denote the group that corresponds to , then the adjoint representation of on can be restricted to and this submanifold is isometrical to the Hilbert space E _ε, where is the decomposition of the twisted loop algebra with respect to the induced involutionρ₀.We thus obtain an affine representation on E _ε and we show that this representation is polar, i. e., there exists a submanifold that intersects all orbits, and intersects them orthogonally. Received: 16 February 2000 RID=" ID="Supported by a DFG grant.     

Lorentzian Clairaut submersions

We investigate a class of semi-Riemannian submersions satisfying a Lorentzian analogue of the classical Clairaut's relation for surfaces of revolution. We show that a Lorentzian submersion with one-dimensional fibers is Clairaut if and only if the fibers are totally umbilic with a gradient field as the normal curvature vector field. We also investigate the behavior of timelike and null geodesics in Lorentzian Clairaut submersions. In particular, every null geodesic of a Lorentzian Clairaut submersion with one-dimensional fibers projects to a pregeodesic in the base space with respect to a conformally related metric on the base space if and only if the integrability tensor of the submersion vanishes.     

Comparison results for the volume of geodesic celestial spheres in Lorentzian manifolds

Volume comparison results are obtained for the volume of geodesic celestial spheres in Lorentzian manifolds and the corresponding objects in Lorentzian space forms. Also, as a rigidity result it is shown that the volume of geodesic celestial spheres is independent of the instantaneous observer if and only if the spacetime has constant curvature.     

Perturbed closed geodesics are periodic orbits: Index and transversality

We study the classical action functional ${\cal S}_V$ on the free loop space of a closed, finite dimensional Riemannian manifold M and the symplectic action on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in M. The potential $V\in C^\infty(M\times S^1,\mathbb{R})$ serves as perturbation and we show that both functionals are Morse for generic V. In this case we prove that the Morse index of a critical point x of equals minus its Conley-Zehnder index when viewed as a critical point of and if is trivial. Otherwise a correction term +1 appears. Received: 21 May 2001; in final form: 10 October 2001 / Published online: 4 April 2002     

The fibre bundle of degenerate tangent planes of a Lorentzian manifold and the smoothness of the null sectional curvature

The Grassmann bundle of degenerate tangent planes to a Lorentzian manifold is introduced and several of its properties are deduced. After a suitable normalization, the null sectional curvature associated to a Lorentzian metric may be considered as a well-defined function on the Grassmann bundle of degenerate tangent planes. The smoothness of this function is proved and some consequences are obtained.     

Analytic integrability and characterization of centers for nilpotent singular points

A method which provides necessary conditions to obtain a local analytic first integral in a neighborhood of a nilpotent singular point is developed. As an application we provide sufficient conditions in order that systems of the form where P_n and Q_n are homogeneous polynomials of degree n = 2, 3, 4, 5 have a local analytic first integral of the form H=y²+F(x, y), where F starts with terms of order higher than 2. We remark that, in general, the existence of such integral is only guaranteed when the singular point is a nilpotent center and the system has a formal first integral, see [6]. Therefore, we characterize the nilpotent centers of systems which have a local analytic first integral.