Invariant f-structures on the flag manifolds SO(n)/SO(2)×SO(n-3)

Invariant structures on homogeneous manifolds are of fundamental importance in differential geometry. Recall that an affinor structure F (i.e., a tensor field F of type (1,1)) on a homogeneous manifold G/H is called invariant (with respect to G) if for any g ∈ G we have dτ(g)◦F = F ◦dτ(g), where τ(g)(xH)= (gx)H . An important place among homogeneous manifolds is occupied by homogeneousΦ-spaces [8, 9] of order k (which are also referred to as k-symmetric spaces [17]), that is, the homogeneous spaces generated by Lie group automorphisms Φ such that Φk = id. Each k-symmetric space has an associated object, the commutative algebra (θ) of canonical affinor structures [7, 8], which is a commutative subalgebra of the algebra of all invariant affinor structures on G/H . In its turn, (θ) contains well-known classical structures, in particular, f -structures in the sense of Yano [19] (i.e., affinor structures F = f satisfying f 3 + f = 0). It should be mentioned that an f -structure compatible with a (pseudo-)Riemannian metric is known to be one of the central objects in the concept of generalized Hermitian geometry [14]. From this point of view it is interesting to consider manifolds of oriented flags of the form