Vol. 310, No. 2, 2021

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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 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
Authors
Miroslava Antić
Faculty of Mathematics
University of Belgrade
Belgrade
Serbia
Haizhong Li
Department of Mathematical Sciences
Tsinghua University
Beijing
China
Luc Vrancken
Laboratoire de Mathématiques pour l’Ingénieur
Université Polytechnique Hauts-de-France
Valenciennes
France
Department of Mathematics
KU Leuven
Leuven
Belgium
Xianfeng Wang
School of Mathematical Sciences and LPMC
Nankai University
Tianjin
China