Every centroaffine Tchebychev hyperovaloid is ellipsoid

Xiuxiu Cheng, Zejun Hu and Luc Vrancken

Vol. 315 (2021), No. 1, 27–44

DOI: 10.2140/pjm.2021.315.27

Abstract

We study locally strongly convex Tchebychev hypersurfaces, namely the centroaffine totally umbilical hypersurfaces, in the $\left(n+1\right)$-dimensional affine space ${ℝ}^{n+1}$. 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 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.

Keywords

centroaffine hypersurface, Tchebychev hypersurface, shape operator, difference tensor, hyperovaloid, ellipsoid.

Mathematical Subject Classification 2010

Primary: 53A15

Secondary: 53C23, 53C24

Milestones

Received: 22 October 2019

Revised: 4 July 2020

Accepted: 14 September 2021

Published: 13 December 2021