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

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 $G$.