On Weakly Reflective PF Submanifolds in Hilbert Spaces

Abstract

We prove that any polar action on a separable Hilbert space by a connected Hilbert Lie group does not have exceptional orbits. This generalizes a result of Berndt, Console and Olmos in the finite dimensional Euclidean case. As an application, we give an alternative proof of the fact that any hyperpolar action on a simply connected compact Riemannian symmetric space by a connected Lie group does not have exceptional orbits.

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

“…(D) Φ −1 (N) is a weakly reflective PF submanifold of V g . The following theorem unifies and extends Theorems 6 and 7 in [18]. In fact, here N need not be homogeneous and the isometry ν ξ need not belong to G×G or Aut(G).…”

“…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].…”

On weakly reflective submanifolds in compact isotropy irreducible Riemannian homogeneous spaces

“…Remark 2. In the previous paper [5,Theorem 8] M was assumed to be a symmetric space of compact type and only the case S = G was considered. Theorem 1 here does not require such assumptions and moreover it characterizes the weakly reflective PF submanifold Φ −1 K (N) precisely.…”

“…Lemma ( [5]). Let M and B be Riemannian Hilbert manifolds, φ : M → B a Riemannian submersion and N a closed submanifold of B.…”

“…Recently the author [5] introduced the concept of weakly reflective submanifolds into a class of proper Fredholm (PF) submanifolds in Hilbert spaces (Terng [10]) and showed that if N is a weakly reflective submanifold of an irreducible Riemannian symmetric space G/K of compact type then its inverse image Φ −1 K (N) under the parallel transport map Φ K : V g → G/K ( [12]) is an infinite dimensional weakly reflective PF submanifold of the Hilbert space V g := L 2 ([0, 1], g) consisting of all L 2paths from [0, 1] to the Lie algebra g of G ([5, Theorem 8]). Using this result many examples of infinite dimensional weakly reflective PF submanifolds were obtained from finite dimensional weakly reflective submanifolds in G/K.…”