  A weakly reflective submanifold is a minimal submanifold of a Riemannian
manifold which has a certain symmetry at each point. In this paper we introduce
this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces
and show that there exist so many infinite dimensional weakly reflective PF
submanifolds in Hilbert spaces. In particular each fiber of the parallel
transport map is shown to be weakly reflective. These imply that in infinite
dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds
which are not totally geodesic, unlike in the finite dimensional Euclidean
case.
