Invariant metric $f$-structures on specific homogeneous reductive spaces

Abstract

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent submodules we establish simple conditions under which an invariant metric f-structure (f,g) belongs to the classes G_1 f, NKf, and Kill f of generalized Hermitian geometry. The statements obtained are then illustrated with four examples. Namely we consider invariant metric f-structures on the manifolds of oriented flags SO(n)/SO(2)\times SO(n-3) (n>=4), the Stiefel manifold SO(4)/SO(2), the complex flag manifold SU(3)/T_{max}, and the quaternionic flag manifold Sp(3)/SU(2)\times SU(2)\times SU(2).