Affine hypersurfaces with constant sectional curvature

Miroslava Antić, Haizhong Li, Luc Vrancken and Xianfeng Wang

Vol. 310 (2021), No. 2, 275–302

DOI: 10.2140/pjm.2021.310.275

Abstract

We use a new approach to study locally strongly convex hypersurfaces with constant sectional curvature in the affine space ${ℝ}^{n+1}$. We prove a nice relation involving the eigenvalues of the shape operator $S$ and the difference tensor $K$ of the affine hypersurface. This is achieved by making full use of the Codazzi equations for both the shape operator and the difference tensor and the Ricci identity in an indirect way. Starting from this relation, we give a classification of locally strongly convex hypersurface with constant sectional curvature whose shape operator $S$ has at most one eigenvalue of multiplicity one.

Keywords

affine hypersurface, affine metric, constant sectional curvature, affine hypersphere

Mathematical Subject Classification

Primary: 53A15

Secondary: 53B20, 53B25

Milestones

Received: 13 May 2020

Revised: 20 November 2020

Accepted: 24 November 2020

Published: 8 March 2021