Every centroaffine Tchebychev hyperovaloid is ellipsoid

TL;DR

This paper proves that the only centroaffine Tchebychev hyperovaloids are ellipsoids, solving a longstanding problem in affine differential geometry by characterizing these hypersurfaces through their Riemannian structure.

Contribution

It establishes that ellipsoids are uniquely characterized as centroaffine Tchebychev hyperovaloids, extending classical theorems to centroaffine differential geometry.

Findings

Ellipsoids are the only centroaffine Tchebychev hyperovaloids.

Characterization of hypersurfaces via Riemannian structures with closed conformal vector fields.

Resolution of a longstanding problem in affine differential geometry.

Abstract

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding problem of trying to generalize the classical theorem of Blaschke and Deicke on affine hyperspheres in equiaffine differential geometry to that in centroaffine differential geometry.