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Marian Hotloś

Publications and source records attributed to Marian Hotloś.

7 recordsLinked to original sources
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.

math.DG↗
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.

math.DG↗
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.

math.DG↗
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.

math.DG↗
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.

math.DG↗
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).

math.DG↗