arXiv subjects

Zbigniew Olszak

Publications and source records attributed to Zbigniew Olszak.

7 recordsLinked to original sources

arXiv2019

The equi-affine curvatures of curves in 3-dimensional pseudo-Riemannian manifolds

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 non-null 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.

math.DG↗

arXiv2018

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

For an arbitrary nondegenerate curve in a pseudo-Riemann\-ian (including Riemannian) 2-manifold, we express the equi-affine curvature with the help of the Frenet (geodesic) curvature of this curve.

math.DG↗

arXiv2015

A note about the torsion of null curves in the $3$-dimensional Minkowski spacetime and the Schwarzian derivative

The main topic of this paper is to show that in the 3-dimensional Minkowski spacetime, the torsion of a null curve is equal to the Schwarzian derivative of a certain function appearing in a description of the curve. As applications, we obtain descriptions of the slant helices, and null curves for which the torsion is of the form $τ=-2λs$, $s$ being the pseudo-arc parameter and $λ=\mathop{\rm constant}\not=0$.

math.DG↗

arXiv2014

The Schouten-van Kampen affine connection adapted to an almost (para) contact metric structure

We study the Schouten-van Kampen connection associated to an almost contact or paracontact metric structure. With the help of such a connection, some classes of almost (para) contact metric manifolds are characterized. Certain curvature properties of this connection are found.

math.DG↗

arXiv2011

On pseudo-Riemannian manifolds with recurrent concircular curvature tensor

It is proved that every concircularly recurrent manifold must be necessarily a recurrent manifold.

math.DG↗

arXiv2011

On 4-dimensional, conformally flat, almost $ε$-Kählerian manifolds

The local structure of 4-dimensional, conformally flat, almost $ε$-Kählerian (i.e., almost pseudo-Kählerian and almost para-Kählerian) manifolds is characterized with the help of left-regular and right-regular paraquaternionic functions. Examples of such structures are discussed.

math.DG↗

arXiv2009

On compact holomorphically pseudosymmetric Kählerian manifolds

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 by W. Jelonek. In his examples, the structure functions change their signs on the manifold.

math.DG↗