Vol. 315, No. 1, 2021

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Every centroaffine Tchebychev hyperovaloid is ellipsoid

Xiuxiu Cheng, Zejun Hu and Luc Vrancken

Vol. 315 (2021), No. 1, 27–44
Abstract

We study locally strongly convex Tchebychev hypersurfaces, namely the centroaffine totally umbilical hypersurfaces, in the (n +1)-dimensional affine space 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
Authors
Xiuxiu Cheng
School of Mathematics and Statistics
Zhengzhou University
Zhengzhou
China
Zejun Hu
School of Mathematics and Statistics
Zhengzhou University
Zhengzhou
China
Luc Vrancken
Laboratoire de Mathématiques pour l’Ingénieur
Université Polytechnique Hauts de France
Valenciennes
France
KU Leuven
Department of Mathematics
Leuven
Belgium