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

Abstract

In the previous paper the author gave a formula for the principal curvatures of proper Fredholm (PF) submanifolds obtained through the parallel transport map over a compact symmetric space and showed an explicit formula for the principal curvatures of orbits of path group actions induced by Hermann actions. In this paper, introducing the concept of canonical isomorphism of path space, we derive a formula for the principal curvatures of PF submanifolds obtained through the parallel transport map over a compact Lie group and show an explicit formula for the principal curvatures of orbits of path group actions induced by sigma-actions. It turns out that all known computational results of principal curvatures of PF submanifolds are special cases of the formula given in the previous paper. Moreover we study austere and weakly reflective properties of orbits of those path group actions and give new examples of infinite dimensional austere PF submanifolds and weakly reflective PF submanifolds in Hilbert spaces.