Zbigniew Olszak

ON THE', EXISTENCE OF GENERALIZED COMPLEX SPACE FORMS

The existence of generalized complex space forms with nonconstant functionh is proved.

On almost cosymplectic (−1, μ, 0)-spaces

AbstractIn 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

On three-dimensional conformally flat quasi-Sasakian manifolds

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ON PARAHOLOMORPHICALLY PSEUDOSYMMETRIC PARA-KÄHLERIAN MANIFOLDS

-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.

The equi-affine and Frenet curvatures of curves in pseudo-Riemannian 2-manifolds

It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold with the same recurrence form.

On compact holomorphically pseudosymmetric Kählerian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function β is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with β = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O1]

On totally geodesic affine immersions

For anyn ≥ 2, we give examples of almost Kähler conformally flat manifoldsM2n which are not Kähler. We discuss the meaning of these examples in the context of the Goldberg conjecture on almost Kahler manifolds.

ON CONTACT METRIC MANIFOLDS

We find necessary and sufficient conditions for a para-Kahlerian manifold to be paraholomorphically pseudosymmetric in terms of the paraholomorphic projective and Bochner curvatures. New examples of such spaces are proposed.