Abstract
In this paper we prove the Magid-Ryan conjecture for 4-dimensional affine hyperspheres in R5. This conjecture states that every affine hypersphere with non-zero Pick invariant and constant sectional curvature is affinely equivalent with either (x12 ±x22)(x32 ±x42...(x2m−12 ±x2m2) = 1 or (x12 ±x22(x32 ±x42)...(x2m−12 ±x2m2)x2m+1 = 1 where the dimensionn satisfiesn = 2m orn =2m + 1. This conjecture was proved in [11] in case the metric is positive definite and in [2] in case the metric is Lorentzian.
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