  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 $G$.
