Ryszard Deszcz

On ricci-pseudosymmetric hypersurfaces in spaces of constant curvature

This article studies the conditions of pseudosymmetry and Ricci-pseudosymmetry, realizedon hypersurfaces of semi-Riemannian spaces of constant curvature. In particular, we derive extrinsic characterizations of pseudosymmetric and Ricci-pseudosymmetric hypersurfaces of semi-Riemannian spaces of constant curvature in terms of the shape operator. As an application, and in the Riemannian case, we extend a theorem by K. Nomizu on semisymmetric hypersurfaces of Euclidean spaces.

Curvature properties of Gödel metric

The main aim of this paper is to investigate the geometric structures admitting by the Godel spacetime which produces a new class of semi-Riemannian manifolds. We also consider some extension of Godel metric.

On pseudosymmetric space–times

An algebraic classification of four‐dimensional pseudosymmetric Einstein space–times is given and more examples of pseudosymmetric space–times are provided.

ON CURVATURE PROPERTIES OF CERTAIN QUASI-EINSTEIN HYPERSURFACES

It is known that the Cartan hypersurfaces of dimension 6, 12 or 24 are non-quasi-Einstein, non-pseudosymmetric, Ricci-pseudosymmetric manifolds. In this paper we investigate quasi-Einstein hypersurfaces in semi-Riemannian space forms satisfying some Walker type identity. Among other things we prove that such hypersurfaces are Ricci-pseudosymmetric manifolds. Using certain result of Magid we construct an example of a quasi-Einstein non-pseudosymmetric Ricci-pseudosymmetric warped product which locally can be realized as a hypersurface in a semi-Riemannian space of constant curvature. In our opinion it is a first example of a hypersurface having the mentioned properties.

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.

On 2-Quasi-Umbilical Hypersurfaces in Conformally Flat Spaces

The main result of this paper states that any 2-quasi-umbilical hypersurface of a semi-Riemannian conformally flat space is a manifold with pseudosymmetric Weyl tensor.

On some family of generalized Einstein metric conditions

We prove that every Einstein manifold of dimension > 4 satisfies some pseudosymmetry type curvature condition. Basing on this fact we introduce a family of curvature conditions. We investigate non-Einstein manifolds satisfying one of these conditions. We prove that every such manifold is pseudosymmetric and satisfies other curvature conditions. We prove also an inverse theorem.

A note on almost kähler manifolds

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.

The compact cyclides of Dupin and a conjecture of B.-Y. Chen

We show that the compact cyclides of Dupin are of infinite type; this result provides further support for a conjecture of Chen.