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Let S3 be the unit hypersphere in the 4-dimensional Euclidean space R4 defined by ∑4 i=1 xi = 1. For each θ with 0 < θ < π/2, we denote by Mθ the Clifford torus in S 3 given by the equations x1 + x 2 2 = cos 2 θ and x3+x 2 4 = sin 2 θ. The Clifford torus Mθ is a flat Riemannian manifold equipped with the metric induced by the inclusion map iθ : Mθ → S3. In this note we prove the following rigidity theorem: If f : Mθ → S3 is an isometric embedding, then there exists an isometry A of S3 such that f = A ◦ iθ. We also show no flat torus with the intrinsic diameter ≤ π is embeddable in S3 except for a Clifford torus.

A rigidity theorem for the Clifford tori in 𝑆³