For an orientable compact and connected hypersurface in the Euclidean space R4 with positive scalar curvature S, the shape operator A and the mean curvature α, it is shown that the inequality 6αdetA≥ α2S+4 |?α|2 implies that the hypersurface is a sphere, where ?α is the gradient of α. A similar characterization is also obtained for spheres the Euclidean space R3 (cf. Theorem 2).
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