Rigidity of an Isometric $SL(3,\mathbb{R})$-Action

TL;DR

This paper characterizes certain pseudo-Riemannian manifolds with dense $SL(3,R)$-actions, showing they are essentially quotients of simple Lie groups, revealing rigidity properties of such geometric structures.

Contribution

It provides a classification of manifolds with dense $SL(3,R)$-actions, demonstrating their isometric relation to simple Lie groups under weak irreducibility.

Findings

Manifolds with dense $SL(3,R)$-actions are quotients of simple Lie groups.

Weak irreducibility implies the manifold is isometric to a simple Lie group or its quotient.

The result characterizes the rigidity of $SL(3,R)$-actions on pseudo-Riemannian manifolds.

Abstract

We characterize the universal covering of connected analytic pseudo-Riemannian manifolds which admit a non-trivial and isometric action of the simple Lie group $SL(3,\mathbb{R})$ with a dense orbit preserving a finite volume. If such manifold is also weakly irreducible we prove that $M$ is isometric to, or a quotient space of, a simple Lie group containing $SL(3,\mathbb{R})$.