A Sufficient Condition for Orbits of Hermann Actions to be Weakly Reflective

Austere and arid properties for PF submanifolds in Hilbert spaces

“…Note that there exist examples of weakly reflective submanifolds which satisfy both conditions (a) and (b): a weakly reflective submanifold M reviewed in Example 5.4 (the case m = 2) satisfies (b) for all n ∈ Z ≥2 and in particular if n is even then it also satisfies (a). Although examples by Ohno [23] and by Kimura-Mashimo [14] satisfy the condition (a), Enoyoshi's example ( [3]) does not satisfy (a) but (b). It should be also noted ( [22]) that under suitable assumptions if M satisfies the condition (b) then its inverse image under the parallel transport map is a weakly reflective PF submanifold satisfying the condition (b).…”

“…It is an interesting problem to study submanifold geometry of orbits under isometric actions of Lie groups and to determine their weakly reflective orbits (e.g. [12], [23], [3]).…”

Geometric Properties of Orbits of Hermann actions

“…However, austere orbits and minimal orbits of Hermann actions is not necessary weakly reflective and arid, respectively. In [15], when (θ 1 θ 2 ) 2 = id, we examine a sufficient condition for orbits of Hermann actions to be weakly reflective in terms of symmetric triad with multiplicities.…”

“…On the assumption θ 1 θ 2 = θ 2 θ 1 , a sufficient condition for orbits of Hermann action to be weakly reflective and a characterization of biharmonic orbits of Hermann actions were given in [15] and [16,17].…”

“…In particular, [15] gives weakly reflective orbits of Hermann action which satisfy θ 1 θ 2 = θ 2 θ 1 .…”