Compact Lorentzian Manifolds Research Articles

On Closed Geodesics in Lorentz Manifolds

We construct compact Lorentz manifolds without closed geodesics.

Isometry group of Lorentz manifolds: A coarse perspective

We prove a structure theorem for the isometry group \({\text {Iso}}(M,g)\) of a compact Lorentz manifold, under the assumption that a closed subgroup has exponential growth. We don’t assume anything about the identity component of \({\text {Iso}}(M,g)\), so that our results apply for discrete isometry groups. We infer a full classification of lattices that can act isometrically on compact Lorentz manifolds. Moreover, without any growth hypothesis, we prove a Tits alternative for discrete subgroups of \({\text {Iso}}(M,g)\).

Spectra of Compact Quotients of the Oscillator Group

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. For every lattice $L$ in ${\rm Osc}_1$, we compute the decomposition of the right regular representation of ${\rm Osc}_1$ on $L^2(L\backslash{\rm Osc}_1)$ into irreducible unitary representations. This decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold $L\backslash {\rm Osc}_1$.

Lorentzian manifolds with causal Killing vector field: causality and geodesic connectedness

We prove that a compact Lorentzian manifold $$({\overline{M}},{\overline{g}})$$ admitting a causal Killing vector field is totally vicious or it contains a compact achronal Killing horizon. In particular a compact spacetime which satisfies the null generic condition and admits a causal Killing vector field is totally vicious. If in addition, its universal Lorentzian covering is globally hyperbolic then it is geodesically connected. In the non-compact case, we prove that a chronological spacetime admitting a complete causal Killing vector field, a smooth spacelike partial Cauchy hypersurface S and satisfying the null generic condition is stably causal. If additionally S is compact then the spacetime is globally hyperbolic. We also determine the geodesic connectedness in this case.

Aubry–Mather theory for Lorentzian manifolds

We introduce a version of Aubry–Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of the Mather’s graph theorem. A class of examples, the Lorentzian Hedlund examples, shows the optimality of the obtained results.

CONFORMAL ACTIONS OF REAL-RANK 1 SIMPLE LIE GROUPS ON PSEUDO-RIEMANNIAN MANIFOLDS

Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be conformally flat, confirming the idea that in a "good" dynamical context, a geometry is determined by its automorphisms group. This completes anterior investigations on pseudo-Riemannian conformal actions of semi-simple Lie groups of maximal real-rank. Combined with these results, we obtain as corollary the list of semi-simple Lie groups without compact factor that can act on compact Lorentzian manifolds. We also derive consequences in CR geometry via the Fefferman fibration.

Lorentzian manifolds with a conformal action of SL(2,R)

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat – that is, everywhere locally conformally diffeomorphic to Minkowski space-time. When the group is non-compact and not locally isomorphic to SO (1,n), n \geq 2 , we derive global conclusions, extending a theorem of [18] to some simple Lie groups of real-rank 1. This result is also a first step towards a classification of conformal groups of compact Lorentzian manifolds, analogous to a classification of their isometry groups due to Adams, Stuck and, independently, Zeghib [1, 2, 32].

On the C k-embedding of Lorentzian manifolds in Ricci-flat spaces

In this paper, we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area and then motivate both the mathematical and physical interests in this problem. We show that any n-dimensional compact Lorentzian manifold (Mn, g), with g in the Sobolev space Hs+3, s>n2, admits an isometric embedding in a (2n + 2)-dimensional Ricci-flat semi-Riemannian manifold. The sharpest result available for these types of embeddings, in the general setting, comes as a corollary of Greene’s remarkable embedding theorems R. Greene [Mem. Am. Math. Soc. 97, 1 (1970)], which guarantee the embedding of a compact n-dimensional semi-Riemannian manifold into an n(n + 5)-dimensional semi-Euclidean space, thereby guaranteeing the embedding into a Ricci-flat space with the same dimension. The theorem presented here improves this corollary in n2 + 3n − 2 codimensions by replacing the Riemann-flat condition with the Ricci-flat one from the beginning. Finally, we will present a corollary of this theorem, which shows that a compact strip in an n-dimensional globally hyperbolic space-time can be embedded in a (2n + 2)-dimensional Ricci-flat semi-Riemannian manifold.

Non locally trivializable CR line bundles over compact Lorentzian CR manifolds

We consider compact CR manifolds of arbitrary CR codimension that satisfy certain geometric conditions in terms of their Levi form. Over these compact CR manifolds, we construct a deformation of the trivial CR line bundle over M which is topologically trivial over M but fails to be even locally CR trivializable over any open subset of M. In particular, our results apply to compact Lorentzian CR manifolds of hypersurface type.

The geodesic completeness of compact Lorentzian manifolds admitting a timelike killing vector field revisited: Two new proofs

Compact Lorentzian manifolds admitting a timelike Killing vector field are shown to be complete by means of two different proofs to the original one.

Essential conformal actions of $${{\mathrm{PSL}}}(2,\mathbf {R})$$ PSL ( 2 , R ) on real-analytic compact Lorentz manifolds

The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least $3$ admitting a conformal essential (i.e. conformal, but not isometric) action of a Lie group locally isomorphic to PSL(2,R). It is established by using a general result of M. Gromov on local isometries of real-analytic $A$-rigid geometric structures. As corollary, we deduce the same conclusion for conformal essential actions of connected semi-simple Lie groups on real-analytic compact Lorentz manifolds. This work is a contribution to the understanding of the Lorentzian version of a question asked by A. Lichnerowicz.

Compact Lorentzian holonomy

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. This property is equivalent to admitting a parallel timelike vector field. We give some applications and derive some properties of the space of all such metrics on a given manifold.

Completeness of compact Lorentzian manifolds with abelian holonomy

We address the problem of finding conditions under which a compact Lorentzian manifold is geodesically complete, a property, which always holds for compact Riemannian manifolds. It is known that a compact Lorentzian manifold is geodesically complete if it is homogeneous, or has constant curvature, or admits a time-like conformal vector field. We consider certain Lorentzian manifolds with Abelian holonomy, which are locally modelled by the so called pp-waves, and which, in general, do not satisfy any of the above conditions. %the condition that their curvature sends vectors that are orthogonal to the vector field to a multiple of the vector field. We show that compact pp-waves are universally covered by a vector space, determine the metric on the universal cover, and prove that they are geodesically complete. Using this, we show that every Ricci-flat compact pp-wave is a plane wave.

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.

Lorentzian compact manifolds: Isometries and geodesics

In this work we investigate families of compact Lorentzian manifolds in dimension four. We show that every lightlike geodesic on such spaces is periodic, while there are closed and non-closed spacelike and timelike geodesics. Also their isometry groups are computed. We also show that there is a non trivial action by isometries of H3(R) on the nilmanifold S1×(Γk∖H3(R)) for Γk, a lattice of H3(R).

Actions of discrete groups on stationary Lorentz manifolds

AbstractWe study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (respectively, lightlike) manifold.

Lorentzian similarity manifolds

Abstract An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝm+2), where G = ℝm+2 ⋊ (O(m+1, 1)×ℝ+). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.

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.

Compact Lorentz manifolds with local symmetry

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, as- pherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity com- ponent, then the local isometry orbits in M are roughly fibers of a fiber bundle. A corollary is that if M has an open, dense, locally homogeneous subset, then M is locally homogeneous.

Isometric Actions of Heisenberg Groups on Compact Lorentz Manifolds

We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions – those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a classification of codimension-one actions, under the assumption they are real-analytic.