Compact pseudo-Riemannian Manifolds Research Articles

On locally homogeneous pseudo-Riemannian compact Einstein manifolds

We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford–Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein Riemannian compact manifolds using T. Kobayashi’s conjecture as a guiding principle.

Conformal actions of higher rank lattices on compact pseudo-Riemannian manifolds

We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group by a result of Zimmer. When the real-rank is maximal, we prove that the manifold is conformally flat. This indicates that a global conclusion similar to that of anterior works of Bader, Nevo and Frances, Zeghib in the case of a Lie group action might be obtained. We also give better estimates for actions of cocompact lattices in exceptional groups. Our work is strongly inspired by the recent breakthrough of Brown, Fisher and Hurtado on Zimmer's conjecture.

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.

On locally homogeneous compact pseudo-Riemannian manifolds

On locally homogeneous compact pseudo-Riemannian manifolds

Conformal Killing Forms on Totally Umbilical Submanifolds

In the present paper we shall study the global existence of conformal Killing forms on compact and orientable submanifolds of pseudo-Riemannian manifolds. In addition, we shall determine exact upper bounds of the dimensions of the vector spaces of these forms on compact and orientable pseudo-Riemannian manifolds.

ABOUT PSEUDO-RIEMANNIAN LICHNEROWICZ CONJECTURE

We construct what we think to be the first known examples of compact pseudo-Riemannian manifolds having an essential group of conformal transformations, and which are not conformally flat. Our examples cover all types (p, q), with 2 ≤ p ≤ q.

Conformal actions of nilpotent groups on pseudo-Riemannian manifolds

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure.

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

It is shown that in every dimension n = 3j + 2, j = 1, 2, 3, . . ., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.

On compact manifolds admitting indefinite metrics with parallel Weyl tensor

Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle.

Isometric actions of simple Lie groups on pseudoRiemannian manifolds

Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m0, n0 are the dimensions of the maximal lightlike subspaces tangent toM and G, respectively, where G carries any bi-invariant metric, then we have n0 iU m0. We study G-actions that satisfy the condition n0 = m0. With no rank restrictions on G, we prove that M has a finite covering
M to which the G-action lifts so that
M is G-equivariantly diffeomorphic to an action on a double coset K\L/?£, as considered in Zimmeri¯s program, with G normal in L (Theorem A). If G has finite center and rankR(G) iÝ 2, then we prove that we can choose
M for which L is semisimple and ?£ is an irreducible lattice (Theorem B). We also prove that our condition n0 = m0 completely characterizes, up to a finite covering, such double coset G-actions (Theorem C). This describes a large family of double coset G-actions and provides a partial positive answer to the conjecture proposed in Zimmeri¯s program.

PseudoRiemannian geometry and actions of simple Lie groups

Let G be a connected noncompact simple Lie group acting isometrically on a connected compact pseudoRiemannian manifold M. Denote with n 0 and m 0 the dimension of the maximal null subspaces tangent to G and M, respectively. Then we always have n 0 ⩽ m 0 . Our main result states that, if n 0 = m 0 , then the G-action is, up to a finite covering, an algebraic action. We use this to obtain a complete characterization of a large family of G-actions, thus providing a partial positive answer to the conjecture proposed in Zimmer's program for pseudoRiemannian manifolds. To cite this article: R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Conformal Actions of Simple Lie Groups on Compact Pseudo-Riemannian Manifolds

Conformal Actions of Simple Lie Groups on Compact Pseudo-Riemannian Manifolds

Geodesic completeness and stability

A (connected) Riemannian manifold M is geodesically complete (i.e. each geodesic may be extended to a geodesic with domain ( −∞, + ∞)) iff, as a metric space under the induced Riemannian distance function, M is Cauchy complete ([12], p. 138). Furthermore, all compact Riemannian manifolds are complete. On the other hand, compact pseudo-Riemannian manifolds exist which are not geodesically complete. For example Fierz and Jost[8] have constructed incomplete metrics of the form 2dx dy + h dy2 on T2. Furthermore, Williams [16] has shown that geodesic completeness may fail to be stable for pseudo-Riemannian manifolds. Here a property is said to be stable if the set of metrics for M with this property is open. This failure of stability may occur for both compact and non-compact manifolds. In particular, Williams has found that M = S1 x S1 = ℝ/(2πZ x 2πZ), which may be given the complete Lorentzian metric g = 2dx dy, also admits Lorentzian metrics gn = 2dx dy+(sin(x)/n) dy2 which are geodesically incomplete yet which for large n are arbitrarily close to g in the Whitney fine Cr topologies. In this example the set S = {(0, y)|0 ≤ y ≤ 2π} represents the image of a closed null geodesic for (M, g) and also for all (M, gn). However, for the metrics (M, gn) this closed null geodesic has affine parametrizations which are incomplete because the velocity vector of this null geodesic returns to a scalar multiple of itself after each complete circuit of S. In general, a closed null geodesic will be complete iff its velocity vector returns exactly to itself after each trip around its image.

Compact Clifford-Klein forms of symmetric spaces - revisited

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for G/H, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applications of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of signature (7, 8) with constant curvature, (ii) there exists a compact quotient of the complex sphere of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the tangential space form of signature (p, q) if and only if p is smaller than the Hurwitz-Radon number of q.