On product affine hyperspheres in R~(n+1)

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space R~(n+1) which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3-and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.