On product affine hyperspheres in ${\mathbb{R}}^{n+1}$

Abstract

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ${\mathbb{R}}^{n+1}$ which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvatures. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, affine hyperspheres of dimensions 3 and 4 with parallel Ricci tensor are also classified.

27 pages. This article has been accepted for publication in SCIENCE CHINA Mathematics