  For a Riemannian submersion from a simple compact Lie group with a
bi-invariant metric, we prove the action of its holonomy group on the fibers is
transitive. As a step towards classifying Riemannian submersions with totally
geodesic fibers, we consider the parameterized surface induced by lifting a
base geodesic to points along a geodesic in a fiber. Such a surface is "doubly
ruled" (it is ruled by horizontal geodesics and also by vertical geodesics).
Its characterizing properties allow us to define "doubly ruled parameterized
surfaces" in any Riemannian manifold, independent of Riemannian submersions. We
initiate a study of the doubly ruled parameterized surfaces in compact Lie
groups and in other symmetric spaces by establishing several rigidity theorems
and by providing several examples with unexpected properties.
