[Paper Review] Homogeneous Lorentz manifolds with simple isometry group

Dave Witte|ArXiv.org|Jul 24, 2000

Geometric Analysis and Curvature Flows12 references6 citations

TL;DR

This paper classifies homogeneous Lorentz manifolds with simple isometry groups, proving that for $G = \mathrm{SO}(1,n)^\circ$ ($n \geq 3$) or $G = \mathrm{SO}(2,n)^\circ$ ($n \geq 3$), any closed subgroup $H$ admitting a $G$-invariant Lorentz metric must have identity component $H^\circ$ conjugate to $\mathrm{SO}(1,n-1)^\circ$ or $\mathrm{SO}(1,n)^\circ$, respectively, under the condition that $\mathrm{Ad}_G H$ has noncompact closure. The result completes a classification of such Lorentz structures in the non-Riemannian case for simple Lie groups.

ABSTRACT

Let H be a closed, noncompact subgroup of a simple Lie group G, such that G/H admits an invariant Lorentz metric. We show that if G = SO(2,n), with n > 2, then the identity component of H is conjugate to the identity component of SO(1,n). Also, if G = SO(1,n), with n > 2, then the identity component of H is conjugate to the identity component of SO(1,n-1).

Motivation & Objective

• To classify homogeneous Lorentz manifolds $G/H$ where $G$ is a simple Lie group with finite center and admits a $G$-invariant Lorentz metric.
• To extend N. Kowalsky's classification by removing the compactness assumption but restricting to almost simple $G$, focusing on the non-Riemannian cases $G \simeq \mathrm{SO}(1,n)$ and $G \simeq \mathrm{SO}(2,n)$.
• To determine the structure of closed subgroups $H \subset G$ such that $G/H$ carries a $G$-invariant Lorentz metric and $\mathrm{Ad}_G H$ has noncompact closure.
• To prove that in these cases, the identity component $H^\circ$ must be conjugate to the standard subgroup $\mathrm{SO}(1,n-1)^\circ$ or $\mathrm{SO}(1,n)^\circ$, respectively.

Proposed method

• Uses representation-theoretic techniques on the Lie algebra $\mathfrak{g} = \mathfrak{so}(1,n)$ or $\mathfrak{so}(2,n)$, analyzing the adjoint action of $H$ on $\mathfrak{g}/\mathfrak{h}$.
• Applies the Iwasawa decomposition $\mathfrak{g} = \mathfrak{k} + \mathfrak{a} + \mathfrak{n}$ to study root space decompositions and weight spaces.
• Analyzes the action of $\mathrm{Ad}_G H$ on $\mathfrak{g}/\mathfrak{h}$ via weight spaces and hyperbolic elements, particularly focusing on the image of $\pi(u)^2$ for $u \in \mathfrak{n}$.
• Employs the condition that an $\mathrm{Ad}_G H$-invariant Minkowski form on $\mathfrak{g}/\mathfrak{h}$ exists if and only if the form is preserved under the action of $\mathrm{Ad}_G H$, reducing the problem to algebraic constraints.
• Uses contradiction arguments based on the structure of root spaces $\mathfrak{g}_\alpha$, $\mathfrak{g}_{\alpha+\beta}$, etc., to rule out non-standard subalgebras.
• Applies the fact that compact subgroups $M$ contain no nontrivial unipotent or hyperbolic elements, which is used to derive contradictions when such elements appear in $W = \mathrm{span}\{(\mathrm{ad}_u)^2 v\}$.

Experimental results

Research questions

• RQ1 For $G = \mathrm{SO}(1,n)^\circ$ with $n \geq 3$, what are the possible closed subgroups $H$ such that $G/H$ admits a $G$-invariant Lorentz metric and $\mathrm{Ad}_G H$ has noncompact closure?
• RQ2 For $G = \mathrm{SO}(2,n)^\circ$ with $n \geq 3$, what are the closed subgroups $H$ admitting a $G$-invariant Lorentz metric and noncompact $\mathrm{Ad}_G H$?
• RQ3 Is the only possibility for such $H$ the standard embedding of $\mathrm{SO}(1,n-1)^\circ$ in $\mathrm{SO}(1,n)^\circ$ or $\mathrm{SO}(1,n)^\circ$ in $\mathrm{SO}(2,n)^\circ$?
• RQ4 Can nonstandard subgroups $H$ (e.g., discrete or non-conjugate to standard subalgebras) arise in the non-Riemannian case for simple $G$?
• RQ5 What algebraic conditions on $\mathfrak{h} \subset \mathfrak{g}$ ensure the existence of an $\mathrm{Ad}_G H$-invariant Minkowski form on $\mathfrak{g}/\mathfrak{h}$?

Key findings

• For $G = \mathrm{SO}(1,n)^\circ$ with $n \geq 3$, any closed subgroup $H$ with noncompact $\mathrm{Ad}_G H$ and a $G$-invariant Lorentz metric on $G/H$ must have $H^\circ$ conjugate to $\mathrm{SO}(1,n-1)^\circ$.
• For $G = \mathrm{SO}(2,n)^\circ$ with $n \geq 3$, any such $H$ must have $H^\circ$ conjugate to $\mathrm{SO}(1,n)^\circ$.
• The only exceptions occur when $n=2$ in the $\mathrm{SO}(1,2)^\circ$ case, where $H$ may be discrete.
• The classification relies on analyzing the action of $\mathrm{Ad}_G H$ on the quotient $\mathfrak{g}/\mathfrak{h}$ via root space decomposition and weight space structures.
• The proof uses contradiction by showing that if $H^\circ$ is not conjugate to the standard subgroup, then $W = \mathrm{span}\{(\mathrm{ad}_u)^2 v\}$ must contain nontrivial hyperbolic or unipotent elements, contradicting the compactness of the maximal compact subgroup $M \subset \mathrm{Ad}_G H$.
• The result confirms that the standard symmetric spaces $\mathrm{SO}(1,n)^\circ/\mathrm{SO}(1,n-1)^\circ$ and $\mathrm{SO}(2,n)^\circ/\mathrm{SO}(1,n)^\circ$ are the only homogeneous Lorentz manifolds with simple isometry group in the non-Riemannian case.