Geometric Properties of Orbits of Hermann actions

TL;DR

This paper studies the geometric properties of orbits generated by Hermann actions, focusing on their second fundamental form and conditions for being weakly reflective or arid submanifolds, without assuming involution commutativity.

Contribution

It provides new insights into the geometry of Hermann action orbits, especially regarding their second fundamental form and conditions for special submanifold properties.

Findings

Computed the second fundamental form of Hermann action orbits

Established sufficient conditions for orbits to be weakly reflective

Identified criteria for orbits to be arid submanifolds

Abstract

In this paper, we investigate properties of orbits of Hermann actions as submanifolds without assuming the commutability of involutions which define Hermann actions. In particular, we compute the second fundamental form of orbits of Hermann action, and give a sufficient condition for orbits of Hermann action to be weakly reflective (resp. arid) submanifolds.