This survey article presents certain results concerning natural left invariant para-Hermitian structures on twisted (especially, semidirect) products of Lie groups.
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For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold.
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In our previous paper, almost cosymplectic (κ, μ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, μ, ν)-space can be $$\mathcal{D}$$ -homothetically deformed to an almost cosymplectic −1, μ′, 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, μ, 0)-speces is established: “models” of such spaces are constructed, and it is noted that a given almost cosymplectic (−1, μ 0)-space is locally isomorphic to a corresponding model. In the case when μ is constant, the models can be constructed on the whole of ℝ2n+1 and it is shown that they are left invariant with respect to Lie group actions.
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An almost cosymplectic (κ,μ,ν)-space is by definition an almost cosymplectic manifold whose structure tensor fields φ, ξ, η, g satisfy a certain special curvature condition (see formula (eq1b)). This condition is invariant with respect to the so-called 𝓓-homothetic transformations of almost cosymplectic structures. For such manifolds, the tensor fields φ, h ($= (1/2)ℒ_{ξ}φ$), A ( = -∇ξ) fulfill a certain system of differential equations. It is proved that the leaves of the canonical foliation of an almost cosymplectic (κ,μ,ν)-space with κ<0 are locally flat Kählerian manifolds. A local characterization of such manifolds is established up to a 𝓓-homothetic transformation of the almost cosymplectic structures.