On quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

On hypersurfaces satisfying conditions determined by the Opozda-Verstraelen affine curvature tensor

“…We refer to [28,29,35,50,57] for examples of affine quasi-umbilical hypersurfaces. It easy to check that if (1.7) holds at a point x ∈ U S then (see, Section 5, eqs.…”

Locally homogeneous affine hyperspheres with constant sectional curvature

“…[3,20,23,24,25]). For general dimension only partial results are known, see [4,5,15] for details. In particular for locally strongly convex homogeneous a‰ne hyperspheres, Sasaki [26] reduced the classification to that of homogeneous convex cones.…”

Characterization of the generalized Calabi composition of affine hyperspheres

“…In particular, he introduced a construction, nowadays called the Calabi composition, which shows how to associate with one (or two) hyperbolic affine hypersphere(s) a new hyperbolic affine hypersphere. Such Calabi composition was later generalized by Dillen and Vrancken [6] systematically to obtain a large class of examples of equiaffine homogeneous affine hypersurfaces, some of which have appeared in the list of the partial classification of equiaffine homogeneous affine hypersurfaces [4,5,10]. Most recently and importantly, Hu-Li-Vrancken [8] considered the inverse construction and obtained characterizations of the Calabi composition of hyperbolic hyperspheres, applying these characterizations they can successfully complete the classification of locally strongly convex affine hypersurfaces with parallel cubic form [9], and little later also that of the Lorentzian case [7].…”