The Magid-Ryan conjecture for 4-dimensional affine spheres

In this paper we prove the Magid-Ryan conjecture for 4-dimensional affine hyperspheres in R-5. This conjecture states that every affine hypersphere with non-zero Pick invariant and constant sectional curvature is affinely equivalent with either (x(1)(2) +/- x(2)(2))(x(3)(2) +/- x(4)(2))...(x(2m-1)(2) +/- x(2m)(2)) = 1 or (x(1)(2) +/- x(2)(2))(x(3)(2) +/- x(4)(2))...(x(2m-1)(2) +/- x(2m)(2))x(2m+1) = 1 where the dimension n satisfies n = 2m or n = 2m + 1. This conjecture was proved in [11] in case the metric is positive definite and in [2] in case the metric is Lorentzian.