Affine hypersurfaces with constant sectional curvature

Abstract

We use a new approach to study locally strongly convex hypersurfaces with constant sectional curvature in the affine space (Formula presented). 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.