Quasi-umbilical affine hypersurfaces congruent to their centre map

TL;DR

This paper investigates a special class of convex affine hypersurfaces that are congruent to their centre map, focusing on cases with a specific shape operator eigenvalue structure, and constructs them from affine hyperspheres.

Contribution

It characterizes and constructs strictly convex affine hypersurfaces congruent to their centre map with a shape operator having two distinct eigenvalues.

Findings

Hypersurfaces are constructed from (n-1)-dimensional affine hyperspheres.

The shape operator has two eigenvalues: one of multiplicity 1, and one of multiplicity n-1.

The study provides a classification for these hypersurfaces.

Abstract

In this paper, we study strictly convex affine hypersurfaces centroaffinely congruent to their centre map, in the case when the shape operator has two distinct eigenvalues: one of multiplicity 1, and one nonzero of multiplicity n-1. We show how to construct them from (n-1)-dimensional affine hyperspheres.