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Ryszard Deszcz

Publications and source records attributed to Ryszard Deszcz.

12 recordsLinked to original sources
arXiv2025

Curvature properties of pseudosymmetry type of some 2-quasi-Einstein manifolds

Let (M,g) be a 2-quasi-Einstein non-conformally flat semi-Riemannian manifold of dimension > 3. We prove that if its Riemann-Christoffel curvature tensor R is a linear combination of some Kulkarni-Nomizu tensors formed by the metric tensor g, the Ricci tensor S and its square S^2, then some pseudosymmetry type curvature conditions are satisfied. Certain non-conformally flat warped product manifolds with 2-dimensional base, and in particular some spacetimes, are such 2-quasi Einstein manifolds.

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arXiv2023

On Wintgen ideal submanifolds satisfying some pseudo-symmetry type curvature conditions

Let M be a Wintgen ideal submanifold of dimension n in a real space form R^{n+m}(k) of dimension (n+m) and of constant curvature k, n > 3, m = 1 or m > 1. Let g, R, Ricc, g /\ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms R^{n+m}(k), n > 3, m = 1 or m > 3, satisfying the following pseudo-symmetry type curvature conditions: (i) R.C and Q(g,R) (resp., Q(g,C), Q(g,g/\Ricc), Q(Ricc,R) or Q(Ricc,g/\Ricc)) are linearly dependent; (ii) C.R and Q(g,R) (resp., Q(g,C), Q(g,g/\Ricc), Q(Ricc,R) or Q(Ricc,g/\Ricc)) are linearly dependent; (iii) R.C - C.R and Q(g,R) (resp., Q(g,C), Q(g,g/\Ricc), Q(Ricc,R) or Q(Ricc,g/\Ricc)) are linearly dependent.

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arXiv2023

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

We investigate hypersurfaces M isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n > 3, such that the operator A^3, where A is the shape operator of M, is a linear combination of the operators A^2 and A and the identity operator Id. The main result states that on the set U of all points of M at which the square of the Ricci operator of M is not a linear combination of the Ricci operator and the identity operator, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor S^2 of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the (0,4)-tensor R.S is on U a linear combination of some Tachibana tensors formed by the tensors g, S and S^2. In particular, if M is a hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space of constant curvature, n > 3, with three distinct principal curvatures and the Ricci operator with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.

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arXiv2023

On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions

The difference tensor R.C-C.R of a semi-Riemannian manifold (M,g), dim M > 3, formed by its Riemannian-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0,6)-Tachibana tensors Q(A,T), where A is a symmetric (0,2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

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arXiv2023

A note on some generalized curvature tensor

For any semi-Riemannian manifold (M,g) we define some generalized curvature tensor as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.

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arXiv2020

On geodesic mappings in particular class of Roter spaces

We determine a particular class of Roter type warped product manifolds. We show that every manifold of that class admits a geodesic mapping onto a some Roter type warped product manifold. Moreover, both geodesically related manifolds are pseudosymmetric of constant type.

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arXiv2020

On hypersurfaces satisfying conditions determined by the Opozda-Verstraelen affine curvature tensor

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this tensor for locally strongly convex hypersurfaces M, n>2, with two distinct affine principal curvatures or with three distinct affine principal curvatures assuming that at least one affine principal curvature has multiplicity 1.

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arXiv2019

Hypersurfaces in space forms satisfying some generalized Einstein metric condition

The difference tensor C.R - R.C of Einstein manifolds, some quasi-Einstein manifolds and Roter type manifolds, of dimension n > 3, satisfy the following curvature condition: (A) C.R - R.C = Q(S,C) - (k /(n-1)) Q(g,C). We investigate hypersurfaces M in space forms N satisfying (A). The main result states that if the tensor C.R - R.C of a non-quasi-Einstein hypersurface M in N is a linear combination of the tensors Q(g,C) and Q(S,C) then (A) holds on M. In the case when M is a quasi-Einstein hypersurface in N and some additional assumptions are satisfied then (A) also holds on M.

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arXiv2015

Curvature properties of some class of warped product manifolds

Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p=2 and the fibre is a semi-Riemannian space of constant curvature, if n is greater or equal to 4, then the (0,6)-tensors R.R - Q(S,R) and C.C of such warped products are proportional to the (0,6)-tensor Q(g,C) and the tensor C is expressed by a linear combination of some Kulkarni-Nomizu products formed from the tensors g and S. Thus these curvature conditions satisfy non-conformally flat non-Einstein warped product spacetimes (p=2, n=4). We also investigate curvature properties of pseudosymmetry type of quasi-Einstein manifolds. In particular, we obtain some curvature property of the Goedel spacetime.

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arXiv2015

On pseudosymmetric manifolds

In the literature, there are two different notions of pseudosymmetric manifolds, one by Chaki [7] and other by Deszcz [16], and there are many papers related to these notions. The object of the present paper is to deduce necessary and sufficient conditions for a Chaki pseudosymmetric [7] (resp. pseudo Ricci symmetric [8]) manifold to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also study the necessary and sufficient conditions for a weakly symmetric [58] (resp. weakly Ricci symmetric [59]) manifold by Tamássy and Binh to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also obtain the reduced form of the defining condition of weakly Ricci symmetric manifolds by Tamássy and Binh [59]. Finally we give some examples to show the independent existence of such types of pseudosymmetry which also ensure the existence of Roter type and generalized Roter type manifolds and the manifolds with recurrent curvature 2-form ([2], [29]) associated to various curvature tensors.

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arXiv2014

Curvature Properties of Gödel metric

The main aim of this article is to investigate the geometric structures admitting by the Gödel spacetime which produces a new class of semi-Riemannian manifolds (see Theorem 4.1 and Theorem 4.5). We also consider some extension of Gödel metric (see Example 4.1).

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