The Canonical Isomorphism between Path Spaces and Principal Curvatures of PF Submanifolds

In this paper we completely make clear the relations among the computational results for the principal curvatures of proper Fredholm (PF) submanifolds in Hilbert spaces. To do this, we define and investigate the canonical isomorphism between path spaces, which is a linear isomorphism between Hilbert spaces, equivariant under the gauge transformations, and compatible with the parallel transport map. Using this isomorphism we also extend the author’s previous results concerning austere PF submanifolds and weakly reflective PF submanifolds in Hilbert spaces and give their concrete examples.