Geometric Properties of Orbits of Hermann actions

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.

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

“…Thus we have dϕ(m 0,ǫ ) = g 0,ǫ and ϕ(m α,ǫ ) = g α,ǫ . Therefore the assertion follows by the general theory of orbits of Hermann actions ( [22], see also [21,Section 3]) and α, ξ = 1 2 α, ξ . If σ is involutive then we have the eigenspace decomposition…”

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

“…In this section we review the submanifold geometry of orbits of Hermann actions. The more detailed facts can be found in Ohno [13] (see also ). Throughout this section M = G/K denotes a symmetric space of compact type and K ′ a symmetric subgroup of G with an involutive automorphism θ ′ .…”