Examples of certain kind of minimal orbits of Hermann actions

Abstract

We give examples of certain kind of minimal orbits of Hermann actions and discuss whether each of the examples is austere.

On the indices of minimal orbits of Hermann actions

“…3 and β i (Z 0 ) = 0 (i ∈ {1, · · · , 3n + 2} \ {n + 1, 2n + 2}). This point Z 0 satisfies the condition (I 1 ) (see Section 4 of [Koi3]).…”

“…Let Z 0 be the point of b defined by β n+1 (Z 0 ) = β 3n+2 (Z 0 ) = π 3 and β i (Z 0 ) = 0 (i ∈ {1, · · · , 3n + 2} \ {n + 1, 3n + 2}). This point Z 0 satisfies the condition (I 1 ) (see Section 4 of [Koi3]).…”

“…Denote by the same symbol ω the connection of F ⊥ ( M ) induced from ω and ∇ ω the linear connection on T ⊥ M associated with ω. Denote by ∇ ⊥ the normal connection of the submanifold M . By imitating the discussion in the proof of Theorem A in[Koi3], we can show that…”

“…Also, denote by h Z 0 (or l) the Lie algebra of L. We showed that M is minimal and that h admits a natural reductive decomposition h = l + m h (see Theorem A in [Koi3] or the proof of Theorem A of this paper). Furthermore, we ([Koi3]) showed that the induced metric on the submanifold M in G/K coincides with the H-invariant metric arising from the restriction cB g | m h ×m h of some constant-multiple cB g of B g to m h × m h if one of the following conditions holds: Koi3]). Here we note that, when G is simple, there exists an inner automorphism ρ of G with ρ(K) = H by Proposition 4.39 of [I2].…”

Examples of austere orbits of the isotropy representations for semisimple pseudo-Riemannian symmetric spaces

“…In particular, any austere submanifolds is a submanifold with vanishing mean curvature vector, which is well-known as the minimal condition in Riemannian geometry. Recently, examples of austere submanifolds were given by using the method of orbits on semisimple Riemannian symmetric spaces ( [8], [7], [9]). In [8], Ikawa-Sakai-Tasaki classified austere orbits (in a sphere) of the isotropy representation for a semisimple Riemannian symmetric space in terms of restricted root system theory.…”