# Isometric actions of simple groups and transverse structures: The integrable normal case - Mathematics > Differential Geometry

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 certaingeometric structures. In particular, we prove that for a $G$-action to beequivalent to one on a space of the form $G\times K\backslash H-\Gamma$, itis necessary and sufficient for the $G$-action to preserve a pseudo-Riemannianmetric and a transverse Riemannian metric to the orbits. A similar resultproves that the $G$-actions on spaces of the form $G\times H-\Gamma$ arecharacterized by preserving transverse parallelisms. By relating our techniquesto the notion of the algebraic hull of an action, we obtain infinitesimal Liealgebra structures on certain geometric manifolds acted upon by $G$.

Author: Raul Quiroga-Barranco

Source: https://arxiv.org/