Isometric actions of simple groups and transverse structures: The integrable normal case

TL;DR

This paper establishes global rigidity results for isometric actions of simple Lie groups, characterizing actions that preserve specific geometric structures and relating them to algebraic and infinitesimal Lie algebra frameworks.

Contribution

It provides necessary and sufficient conditions for actions to be equivalent to certain homogeneous models based on preserved geometric structures.

Findings

Actions preserving pseudo-Riemannian and transverse Riemannian metrics are characterized.

Actions on certain spaces are characterized by preserving transverse parallelisms.

Infinitesimal Lie algebra structures are derived from the algebraic hull of the actions.

Abstract

For actions with a dense orbit of a connected noncompact simple Lie group $G$, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a $G$-action to be equivalent to one on a space of the form $(G\times K\backslash H)/\Gamma$, it is necessary and sufficient for the $G$-action to preserve a pseudo-Riemannian metric and a transverse Riemannian metric to the orbits. A similar result proves that the $G$-actions on spaces of the form $(G\times H)/\Gamma$ are characterized by preserving transverse parallelisms. By relating our techniques to the notion of the algebraic hull of an action, we obtain infinitesimal Lie algebra structures on certain geometric manifolds acted upon by $G$.