The Geometry of Orbits of Hermann Type Actions

Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups

“…First, we consider orbits of commutative Hermann actions. Since all orbits of Hermann actions satisfy ∇ ⊥ X τ 1 H = 0 (see [IST1]), we can apply Theorem 2.1.…”

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

“…From this section, we apply the results in Section 3 to the orbits of Hermann actions using symmetric triads (cf. [10], [11], [12]), and determine biharmonic regular orbits of cohomogeneity one Hermann actions. For this purpose, we express the tension field and the square norm of the second fundamental form of orbits of Hermann actions in terms of symmetric triads (Theorem 4.8).…”

“…Then, (N 1 , ·, · ) and (N 2 , ·, · ) are Einstein manifolds with Einstein constant c = 1/2. It is known that the tension field of an orbit of a Hermann action is parallel in the normal bundle (see [12]), i.e. ∇ ⊥ X τ H = 0 for every vector field X on the orbit K 2 π 1 (x).…”

Double Satake diagrams and canonical forms in compact symmetric triads

“…In fact, in the case when (G, θ 1 , θ 2 ) is commutable, its canonical form is given by a commutative compact symmetric triad (G, θ ′ 1 , θ ′ 2 ) ∼ (G, θ 1 , θ 2 ). Many mathematicians contributed to study the Hermann action of such (G, θ ′ 1 , θ ′ 2 ) (for example, [10], [11], [17], [18], [20]). On the other hand, the study of Hermann actions for non-commutable case was found in [5] and [19].…”