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

TL;DR

This paper derives explicit formulas for principal curvatures of orbits under path group actions induced by Hermann actions on symmetric spaces, and investigates conditions for these orbits to be austere, revealing new infinite-dimensional austere submanifolds.

Contribution

It provides explicit curvature formulas and criteria for austere orbits in the context of path group actions induced by Hermann actions, extending previous results to infinite dimensions.

Findings

Explicit principal curvature formulas for Hermann action orbits

Conditions for orbits to be austere submanifolds

Existence of infinite-dimensional austere submanifolds in Hilbert spaces

Abstract

It is known that an isometric action of a Lie group on a compact symmetric space gives rise to a proper Fredholm action of a path group on a path space via the gauge transformations. In this paper, supposing that the isometric action is a Hermann action (i.e. an isometric action of a symmetric subgroup of the isometry group) we give an explicit formula for the principal curvatures of orbits of the path group action and study the condition for those orbits to be austere, that is, the set of principal curvatures in the direction of each normal vector is invariant under the multiplication by minus one. To prove the results we essentially use the facts that Hermann actions are hyperpolar and all orbits of Hermann actions are curvature-adapted submanifolds. The results greatly extend the author's previous result in the case of the standard sphere and show that there exist a larger number of infinite dimensional austere submanifolds in Hilbert spaces.