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Locally conformally homogeneous Lorentzian spaces

Thomas Leistner, Lilia Mehidi, Abdelghani Zeghib

TL;DR

The paper classifies locally conformally homogeneous Lorentzian manifolds of dimension at least $3$ with essential local conformal transformations, proving they are either conformally flat or locally conformally equivalent to a homogeneous plane wave. It develops a framework based on Gromov’s rigidity and algebraic isotropy to derive a parabolic (null) isotropy reduction, yielding canonical lightlike distributions and a Heisenberg-type Killing algebra that acts on leaves of a null foliation. This leads to a precise plane-wave structure in the non-conformally-flat case and shows that such plane waves are realized as Penrose limits within the conformal class. The results unify and extend previous work on globally homogeneous cases, provide a conceptual route to plane waves in the locally homogeneous setting, and establish Penrose-limit invariance within conformal geometry, linking rigid geometric structures to gravitational limits.

Abstract

We study locally conformally homogeneous Lorentzian manifolds of dimension at least $3$, admitting an essential pseudo-group of local conformal transformations. Generalizing a recent result of Alekseevsky and Galaev, we show that any such manifold $(M,g)$ is either conformally flat, or locally conformally equivalent to a homogeneous plane wave. When the manifold is non-conformally flat, we show the existence of a codimension-one lightlike foliation of Heisenberg type, which leads to the plane wave structure. Our approach relies on tools from Gromov's theory of rigid transformations. Finally, we observe that the plane wave metric in the conformal class coincides with the Penrose limit of $(M,g)$ along some null geodesic.

Locally conformally homogeneous Lorentzian spaces

TL;DR

The paper classifies locally conformally homogeneous Lorentzian manifolds of dimension at least with essential local conformal transformations, proving they are either conformally flat or locally conformally equivalent to a homogeneous plane wave. It develops a framework based on Gromov’s rigidity and algebraic isotropy to derive a parabolic (null) isotropy reduction, yielding canonical lightlike distributions and a Heisenberg-type Killing algebra that acts on leaves of a null foliation. This leads to a precise plane-wave structure in the non-conformally-flat case and shows that such plane waves are realized as Penrose limits within the conformal class. The results unify and extend previous work on globally homogeneous cases, provide a conceptual route to plane waves in the locally homogeneous setting, and establish Penrose-limit invariance within conformal geometry, linking rigid geometric structures to gravitational limits.

Abstract

We study locally conformally homogeneous Lorentzian manifolds of dimension at least , admitting an essential pseudo-group of local conformal transformations. Generalizing a recent result of Alekseevsky and Galaev, we show that any such manifold is either conformally flat, or locally conformally equivalent to a homogeneous plane wave. When the manifold is non-conformally flat, we show the existence of a codimension-one lightlike foliation of Heisenberg type, which leads to the plane wave structure. Our approach relies on tools from Gromov's theory of rigid transformations. Finally, we observe that the plane wave metric in the conformal class coincides with the Penrose limit of along some null geodesic.
Paper Structure (23 sections, 22 theorems, 76 equations)

This paper contains 23 sections, 22 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Terminology
  3. Plane waves
  4. Basic properties of plane waves
  5. Determining a plane wave by its Killing fields
  6. Isotropy group and reductions
  7. The isotropy
  8. Reduction to parabolic isotropy
  9. The simply connected case
  10. The $B^t$-adjoint-action on ${\mathfrak{g}}$
  11. Canonical ${\mathfrak{g}}$-invariant lightlike distributions
  12. Towards a Heisenberg algebra of Killing vector fields
  13. Case ${\mathfrak{h}} \subset {\mathfrak{g}}^0 \oplus {\mathfrak{g}}^1$
  14. Case ${\mathfrak{h}} \subset {\mathfrak{g}}^0 \oplus {\mathfrak{g}}^{-1}$
  15. Proof of Proposition \ref{['Prop. existence of foliation']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\bf P$ be its pseudo-group of local conformal transformations, that is the collection of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e. $M$ is an orbit of $\bf P$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3: Jordan decomposition in algebraic groups
  • Corollary 4.4
  • ...and 30 more