INVARIANT METRIC f-STRUCTURES ON SPECIFIC HOMOGENEOUS REDUCTIVE SPACES

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)× 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)×SU(2)× SU(2). Introduction The concept of generalized Hermitian geometry (see, for example, [16]) was created in the 1980s as a natural consequence of the development of Hermitian geometry and the theory of almost contact structures. One of the central objects in this concept is the metric f -structure (f, g), that is, an f -structure [21] f compatible with an invariant Riemannian metric g. An interesting problem that arises in this context is to determine whether a given metric f -structure belongs to the main classes of generalized Hermitian geometry, for example, to the classes G1f (see [16]), NKf (see [6] and [7]), and Kill f (see [14] and [15]). It should be emphasized that in the case of naturally reductive manifolds [18] there exist a number of results that transform this problem into an easy computational task ([6], [8], [4], [5]). However, in the case of an arbitrary Riemannian metric this problem is not an easy one, at least because it involves the calculation of the implicitly defined Levi-Civita connection. In this paper we consider invariant metric f -structures (f, g) on specific homogeneous reductive spaces G/H , namely on homogeneous reductive spaces that satisfy the following set of conditions: 1) G is a compact semisimple Lie group (hence the Killing form B of G is negative definite). 2) The reductive complement m admits the decomposition m = m1 ⊕ m2 ⊕ m3 2000 Mathematics Subject Classification. Primary 53C15, 53C30; Secondary 53C10.