Austere and arid properties for PF submanifolds in Hilbert spaces

Abstract

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.2010 Mathematics Subject Classification. 53C40.

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

“…In [19] and [21] the author gave a formula for the principal curvatures of PF submanifolds obtained through the parallel transport map Φ K over a compact symmetric space G/K. In this section, from that formula we derive a formula for the principal curvatures of PF submanifolds obtained through the parallel transport map Φ over a compact Lie group.…”

“…Afterwards Koike [14] gave a formula for the principal curvatures of Φ −1 K (N) for a curvature-adapted submanifold N of a compact symmetric space G/K. The author ( [19], [21]) corrected and refined that formula and derived an explicit formula for the principal curvatures of P (G, H × K)-orbits where H is a symmetric subgroup of G. These together with the results in this paper are summarized in the following diagram. Here right squigarrows mean that right sides can be thought of special cases of left sides by Theorem 3.2.…”

“…Fibers of Φ K (The author [19]) Fibers of Φ (King-Terng [13]) ⇑ ⇑ PF submanifold Φ −1 K (N) (Koike [14], the author [19], [21])…”

Curvatures and austere property of orbits of path group actions induced by Hermann actions

“…In [8] and [11] an explicit formula for principal curvatures of PF submanifolds was given under the assumption that PF submanifolds are inverse images of curvature adapted submanifolds in compact symmetric spaces under the parallel transport map. In this section we refine that formula to the case of c-curvature adapted submanifolds so that orbits of Hermann actions can be applied.…”

“…In Section 3 we formulate the curvature adapted property of orbits of Hermann actions. In Section 4 we refine the formula for principal curvatures of PF submanifolds obtained through the parallel transport map, which was originally computed in [8] and [11]. Using this formula, in Section 5 we derive an explicit formula for orbits of P (G, K ′ × K)-actions.…”