Weakly reflective submanifolds and austere submanifolds

Abstract

An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by À1. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.

Austere and arid properties for PF submanifolds in Hilbert spaces

“…In the finite dimensional case minimal orbits of the isotropy representation of symmetric spaces have been systematically studied (e.g. [11], [12]). It may be interesting to ask whether there are similar properties for minimal orbits in the isotropy representations of Kac-Moody symmetric spaces.…”

“…In [12] Ikawa Sakai and Tasaki introduced a certain kind of austere submanifold which has a global symmetry, which they call a weakly reflective submanifold. A submanifold M immersed in a finite dimensional Riemannian manifoldM is called weakly reflective if for each normal vector ξ at each p ∈ M there exists an isometry ν ξ ofM which satisfies ν ξ (p) = p, dν ξ (ξ) = −ξ, ν ξ (M) = M.…”

Hamiltonian minimality of normal bundles over the isoparametric submanifolds

“…Moreover, we have m ξ = m η (cf. [13]). Since the Weyl group acts on the root system, 2η ∈ R whenever 2ξ ∈ R, and these length are also same.…”

“…It is known that there exist a unique austere hypersurface in the family of parallel hypersurfaces of an isoparametric hypersurface N when the multiplicities of the principal curvatures are same. Moreover, the austere orbits of s-representations are classified in [13] (see also [6]). Thus, these twisted normal cones give examples of special Lagrangian varieties.…”

“…First we recall some general arguments for orbits of s-representation. We refer to [14] and [27] (see also [11] and [13]). …”