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

Abstract

It is known that an isometric action of a Lie group on a compact symmetric space induces a proper Fredholm action of a path group on a path space via the gauge transformations. In this paper, supposing that the isometric action is a Hermann action, we study the principal curvatures of orbits of the induced path group action.

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…”

“…In [19] and [21] the author gave a formula for the principal curvatures of PF submanifolds obtained through the parallel transport map Φ K over a compact symmetric space G/K. In this section, from that formula we derive a formula for the principal curvatures of PF submanifolds obtained through the parallel transport map Φ over a compact Lie group.…”

“…Let G be a connected compact semisimple Lie group with a bi-invariant Riemannian metric, Φ : V g → G the parallel transport map and N a submanifold of G. Suppose that N is k-curvature-adapted ( [21]), that is, for each a ∈ G the following conditions are satisfied: (i) for every v ∈ T ⊥ a N the Jacobi operator R v leaves T a N invariant, (ii) for each v ∈ T ⊥ a N there exists a k-dimensional abelian subspace t in g satisfying v ∈ dl a (t) ⊂ T ⊥ a N such that the union…”

“…In [21] the author showed an explicit formula for the principal curvatures of orbits of P (G, H × K)-actions and showed the following theorem. Here a submanifold is called austere ( [5]) if the set of principal curvatures with their multiplicities in the direction of each normal vector is invariant under the multiplication by (−1).…”

“…Theorem ( [21]). Let G/K be a symmetric space of compact type and H a symmetric subgroup of G. Suppose that G is simple.…”