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On Kuiper's conjecture

Thomas Cecil·Quo-Shin Chi·Gary Jensen
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Abstract

We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifold M in $\R^n$, whose tube $M_ε$ satisfies this finiteness condition, is analytic algebraic and is a connected component of an irreducible algebraic set. In particular, we prove that every taut submanifold of dimension $m \leq 4$ is algebraic.

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Thomas Cecil, Quo-Shin Chi, Gary Jensen. 2007-07-31. On Kuiper's conjecture. https://arxiv.org/abs/math/0512089

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