Homogeneous Hermitianf-manifolds

Generalized symmetric spaces, Yu. P. Solovyov’s formula, and the generalized Hermitian geometry

“…It should be mentioned that the result of Corollary 6 for k = 4 was already obtained in [6] (for a naturally reductive metric) and [16] (for the general case). We also note that Riemannian 4-symmetric spaces of classical compact Lie groups were classified in [31].…”

“…By Theorem 5, it follows (see [6]) that any canonical metric f -structure on a naturally reductive k-symmetric space (G/H, g) is a G 1 f -structure, and any canonical almost Hermitian structure J is a G 1 -structure. It gives a wealth of invariant examples of the structures of such a kind for the generalized Hermitian geometry, in particular, for Hermitian geometry.…”

“…Therefore, this gives a long list of homogeneous Hermitian f -manifolds of semisimple type, whose f -structure is generally nonintegrable. For homogeneous spaces of solvable type, the 6-dimensional generalized Heisenberg group (N, g) represented as a Riemannian homogeneous 4-symmetric space (see [57]) is a homogeneous Hermitian f -manifold with respect to the canonical f -structure f = 1 2 (θ−θ 3 ) (see [6]), where the Riemannian metric g is not naturally reductive.…”

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

“…It should be mentioned that if G/H is a regular Φ-space, G a semisimple Lie group then G/H is a naturally reductive space with respect to the (pseudo-)Riemannian metric g induced by the Killing form of the Lie algebra g (see [17]). In [1], [2], [3] and [4] a number of results helpful in checking whether the particular f -structure on a naturally reductive space belongs to the main classes of generalized Hermitian geometry was obtained.…”