On weakly reflective submanifolds in compact isotropy irreducible Riemannian homogeneous spaces

Abstract

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.2010 Mathematics Subject Classification. 53C40.

On the geometry of orbits of path group actions induced by sigma-actions

“…By definition weakly reflective submanifolds are austere submanifolds. The author [18] extended the concept of weakly reflective submanifolds to a class of PF submanifolds in Hilbert spaces and studied the weakly reflective property of PF submanifolds obtained through the parallel transport map (see also [20]). Considering the canonical isomorphism of path space we will prove the following theorem (Theorem 8.1) which unifies and extends some results in [18].…”