  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.
