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

TL;DR

This paper investigates curvature conditions of hypersurfaces in affine space using the Opozda-Verstraelen tensor, focusing on locally strongly convex hypersurfaces with specific principal curvature multiplicities.

Contribution

It characterizes pseudosymmetry-type curvature conditions for hypersurfaces with distinct affine principal curvatures using the Opozda-Verstraelen tensor.

Findings

Curvature conditions for hypersurfaces with two distinct affine principal curvatures.

Curvature conditions for hypersurfaces with three distinct affine principal curvatures.

Results assuming at least one principal curvature has multiplicity one.

Abstract

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.