  In 1995, C.-L. Terng associated to each hyperpolar action on a compact
symmetric space, a hyperpolar proper Fredholm (PF) action on a Hilbert space.
This is a group action by an infinite dimensional path group and it acts on a
Hilbert space via the gauge transformations. Those two hyperpolar actions are
related through an equivariant Riemannian submersion called the parallel
transport map and they have close relations to the infinite dimensional
symmetric spaces called affine Kac-Moody symmetric spaces. In this paper we
define a linear isomorphism between Hilbert spaces and show that it is
equivariant with respect to the gauge transformations and is compatible with
the parallel transport map. Using this isomorphism we extend and unify all
known computational results of principal curvatures of PF submanifolds in
Hilbert spaces. Especially we study the submanifold geometry of orbits of
hyperpolar PF actions associated to sigma-actions and give new examples of
austere PF submanifolds in Hilbert spaces. Moreover we show that the
isomorphism between Hilbert spaces given here corresponds to a natural
isomorphism between affine Kac-Moody symmetric spaces of group type.
