Invariant metric 𝑓-structures on specific homogeneous reductive spaces

Abstract

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum 𝓂 = 𝓂₁⊕𝓂₂⊕𝓂3 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 G₁f , 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)x SO(n-3)(n ¸≥ 4), the Stiefel manifold SO(4)/SO(2), the complex flag manifold SU(3)/𝑇ₘₐₓ, and the quaternionic flag manifold Sp(3)/SU(2) x SU(2) x SU(2).