We find geometric conditions on a four-dimensional Hermitian manifold endowed with metric connection totally skew-symmetric torsion under which the complex structure is harmonic map from into its twistor space considered natural family of Riemannian metrics defined by means and given base manifold.

In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients Γ﹨G, where G is an almost nilpotent Lie group whose nilradical has one-dimensional commutator Γ a lattice G. We first obtain characterization Hermitian algebras g n classification result in real dimension six. Then, study ones admitting SKT examine behaviour such under fl...

An almost para-CR structure on a manifold M is given by a distribution HM ⊂ TM together with a field K ∈ Γ(End(HM)) of involutive endomorphisms of HM . If K satisfies an integrability condition, then (HM,K) is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given...

We discuss the alternative algebraic structures on the manifold of quantum states arising from alternative Hermitian structures associated with quantum bi-Hamiltonian systems. We also consider the consequences at the level of the Heisenberg picture in terms of deformations of the associative product on the space of observables.

The pseudo-Riemannian manifold M = (M, g), n ≥ 2 is paraquaternionic Kähler if hol(M) ⊂ sp(n,R)⊕sp(1, R). If hol(M) ⊂ sp(n, R), than the manifold M is called para-hyperKähler. The other possible definitions of these manifolds use certain parallel para-quaternionic structures in End(TM), similarly to the quaternionic case. In order to relate these different definitions we study para-quaternionic...