Papers by Zbigniew Olszak
Commentationes Mathematicae, 1987
Z b ig n ie w O lszak (Wroclaw) Curvature properties of four-dimensional Hermitian manifolds Abst... more Z b ig n ie w O lszak (Wroclaw) Curvature properties of four-dimensional Hermitian manifolds Abstract. In the paper we study Hermitian manifolds of dimension 4. We^obtain basic identities for the Riemann curvature tensors, the Ricci curvature tensors and the scalar curvatures of such manifolds. Then certain sufficient conditions for a 4-dimensional Hermitian manifold to be Kahlerian are derived. Also two examples of Hermitian structures are given, one of them is not locally conformal Kahlerian and the other is flat, globally conformal Kâhlerian and not Kahlerian. 170 Z. Olszak (1.3) 2VX(J)Y = g(X, Y)JB-g(X, JY)B-m(Y)JX + co(JY)X. Define Ф to be the tensor field on M given by (1.4) Ф(Х, Y)=-2oo(Vx{J)Y). As a consequence of (1.3) and (1.4), one obtains that Ф is a 2-form on M and (1.5) Ф = \B]2Q-w л 9, where 9-vooJ. Let АппФ be the annihilator of the form Ф, i.e., it is the distribution Af э ш и А п п Ф (т), where A nnФ(m) = {XeTmM\ Ф(Х, Г) = 0 for ali YeTmM). Consider also another distribution on M. Namely, let К (M) be the Kâhler-nullity distribution М э т и Х (т) (see [2]), where K(m) = {XeTmM\ F* (J) У = 0 for all YeTmM}. With the help of equalities (1.3)-(1.5) the following proposition can be proved. The proof goes just as the proof of Proposition 1.1 in [7]. P roposition 1.2. Let M be a 4-dimensional Hermitian manifold. Then K (M) = Ann Ф and, moreover, (a) the singular points o f К (M) are the vanishing points of со and at such a point K(m) = TmM,
This survey article presents certain results concerning natural left invariant para-Hermitian str... more This survey article presents certain results concerning natural left invariant para-Hermitian structures on twisted (especially, semidirect) products of Lie groups. 1. Preliminaries. Let M be a connected C ∞-differentiable manifold of even dimension 2n. All of the objects involved on M are of class C ∞ too. By X(M) we denote the Lie algebra of vector fields on M. By an almost paracomplex structure on M we mean a (1, 1)-tensor field J on M such that J 2 = Id (the identity tensor field), and the eigendistributions T ± M associated to the eigenvalues ±1 of J, have the same dimension ([8], [6]). The manifold M admits an almost paracomplex structure if and only if there exists a G-structure on M with structural group GL(n, R) × GL(n, R). Let J be an almost paracomplex structure on M and N J be the Nijenhuis torsion tensor field of J,
Tensor. New series, Dec 1, 1995
Colloquium Mathematicum, 2003
It is proved that there exists a non-semisymmetric pseudosymmetric Kähler manifold of dimension 4.
Colloquium Mathematicum, 1987
Acta Mathematica Hungarica, Apr 3, 2012
It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold... more It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold with the same recurrence form.
Banach Center Publications, 2005
Colloquium Mathematicum, 1987
Colloquium Mathematicum, 1979
Periodica Mathematica Hungarica, Oct 1, 1996
Let M be a S-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure fun... more Let M be a S-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function /3 is defined. With the help of this fuuction, we find necessary and sufficient conditions for hf to be conformally flat. Next it is proved that if M is additionally conformally flat with p = const., then (a) A/r is locally a product of R and a P-dimensional Ktilerian 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 J-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [Ol] 1. Introduction An almost contact metric manifold M and its almost contact metric structure (4, I, 7, g) are said to be quasi-Sasakian if the structure is normal and the funclamental a-form @ is closed. The notion of quasi-Sasakian structures was introduced and the first examples were given by Blair [BI]. The simplest examples of quasi-Sasakian structures are those being cosymplectic (i.e., normal with dq = 0. and d@ = 0; here rank 71 = 1) as well as Sa.sa.kian (i.e., normal with dq = +, here za.m& T) = 2n + 1 = dim hf). Homogeneous quasi-Sasakian structnres on the m EMsenberg groups were constructed by Gonzalez and C!binea [GC]. Products of (quasi-)Sasakian manifolds and KLhlerian manifolds a.re also quasirsasakian. Certain sufficient conditions for a quasi-Sasakian manifold to be locally such a product are studied by Blair [Bl], Tanno [T] and Kaaemaki (Kal, Kaz]. It is also known that D-homathetic and homothetic deformatiions of (quasi-)Sasakian structures lead to quasi-Sasakian structures (cf. the author r&J).
Kodai Mathematical Journal, 1981
The purpose of the present paper is to study the structure of almost cosymplectic manifolds. §2 p... more The purpose of the present paper is to study the structure of almost cosymplectic manifolds. §2 presents the basic definitions and some preliminary properties of an almost cosymplectic structure. Examples of such structures are given in § 3. In § 4 we state many important curvature identities for almost cosymplectic manifolds. In § 5 we give certain sufficient conditions for an almost contact metncstructure to be almost cosymplectic. Basing on the identities from §4 we prove in §6 that almost cosymplectic manifolds of non-zero constant sectional curvature do not exist in dimensions greater than three. However it is known (cf. [2]) that such manifolds of zero sectional curvature (i. e. locally flat) exist and they are cosymplectic. Moreover we give certain restrictions on the scalar curvature of almost cosymplectic manifolds which are conformally flat or of constant ^-sectional curvature. All manifolds considered in this paper are assumed to be connected and of class C°°. All tensor fields, including differential forms, are of class C°°. The notation and terminology will be the same as that employed in [2]. § 2. Preliminaries. Let (M, φ, ξ, 77, g) be a (2n+l)-dimensional almost contact metric manifold, that is, Mis a differentiate manifold and {φ, ξ, η, g) an almost contact metric structure on M, formed by tensor fields φ, ξ, η of type (1, 1), (1, 0), (0, 1), respectively, and a Riemannian metric g such that φξ=O, η°φ=0, η(X)=g(X, ξ), g(φX, φY)=g(X, Y)-η(X)η{Y). On such manifold we may always define a 2-form Φ by Φ(X, Y)~g(φX, Y). (Λf, φ, ξ, 7], g) is said to be an almost cosymplectic manifold (cf. [2]) if the forms Φ and Ύ] are closed, i.e. dΦ=0 and dη^O, where d is the operator of exterior differentiation. In particular, if the almost contact structure of an almost cosymplectic manifold is normal, then it is said to be a cosymplectic manifold (cf. [1]).
... DOI No: 10.1142/9789812817976_0020. Source: GEOMETRY AND TOPOLOGY OF SUBMANIFOLDS IX (pp 198-... more ... DOI No: 10.1142/9789812817976_0020. Source: GEOMETRY AND TOPOLOGY OF SUBMANIFOLDS IX (pp 198-207). Author(s): ZBIGNIEW OLSZAK Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50370 Wrocław, Poland. ...
Acta Mathematica Hungarica, Dec 1, 2006
At first, a necessary and sufficient condition for a Kähler-Norden manifold to be holomorphic Ein... more At first, a necessary and sufficient condition for a Kähler-Norden manifold to be holomorphic Einstein is found. Next, it is shown that the socalled (real) generalized Einstein conditions for Kähler-Norden manifolds are not essential since the scalar curvature of such manifolds is constant. In this context, we study generalized holomorphic Einstein conditions. Using the one-to-one correspondence between Kähler-Norden structures and holomorphic Riemannian metrics, we establish necessary and sufficient conditions for Kähler-Norden manifolds to satisfy the generalized holomorphic Einstein conditions. And a class of new examples of such manifolds is presented. Finally, in virtue of the obtained results, we mention that Theorems 1 and 2 of H. Kim and J. Kim [10] are not true in general.
Colloquium Mathematicum, 2017
For an arbitrary nondegenerate curve in a pseudo-Riemannian (including Riemannian) 2-manifold, we... more For an arbitrary nondegenerate curve in a pseudo-Riemannian (including Riemannian) 2-manifold, we express the equi-affine curvature with the help of the Frenet (geodesic) curvature of this curve.
Journal of The Korean Mathematical Society, Jul 31, 2008
Annales Polonici Mathematici, 1986
Tohoku Mathematical Journal, 1979
arXiv (Cornell University), Oct 30, 2019
In this paper, the Cartan frames and the equi-affine curvatures are described with the help of th... more In this paper, the Cartan frames and the equi-affine curvatures are described with the help of the Frenet frames and the Frenet curvatures of a nonnull and non-degenerate curve in a 3-dimensional pseudo-Riemannian manifold. The constancy of the Frenet curvatures of such a curve always implies the constancy of the equi-affine curvatures. We show that the converse statement does not hold in general. Finally, we study the equi-affine curvatures of null curves in 3-dimensional Lorentzian manifolds, and prove that they are related to their pseudo-torsion.
At flrst, a necessary and su-cient condition for a Kahler{Norden manifold to be holomorphic Einst... more At flrst, a necessary and su-cient condition for a Kahler{Norden manifold to be holomorphic Einstein is found. Next, it is shown that the so- called (real) generalized Einstein conditions for Kahler{Norden manifolds are not essential since the scalar curvature of such manifolds is constant. In this con- text, we study generalized holomorphic Einstein conditions. Using the one-to-one correspondence between Kahler{Norden structures and holomorphic Riemannian metrics, we establish necessary and su-cient conditions for Kahler{Norden man- ifolds to satisfy the generalized holomorphic Einstein conditions. And a class of new examples of such manifolds is presented. Finally, in virtue of the obtained results, we mention that Theorems 1 and 2 of H. Kim and J. Kim (10) are not true in general.
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Papers by Zbigniew Olszak