Centroaffine T-Umbilical Hypersurfaces and Pseudo-Parallel Cubic Form.

In this article, we introduce the notion of locally strongly convex centroaffine T-umbilical hypersurfaces, which may be the “simplest” centroaffine hypersurfaces next to the hyperquadrics centered at the origin. These hypersurfaces are proved to be of a conformally flat, quasi-Einstein centroaffine metric and the pseudo-parallel cubic form relative to the Levi-Civita connection of the centroaffine metric. As the main results, we find a geometric characterization of those hypersurfaces as not being of constant sectional curvature, and further classify these hypersurfaces. Meanwhile, for locally strongly convex centroaffine hypersurfaces with pseudo-parallel cubic form, we completely determine the 3-dimensional case, and all dimensions for the conformally flat case. [ABSTRACT FROM AUTHOR]