We prove that any connected proper Dupin hypersurface in Rn is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. From this we also prove that every taut submanifold of dimension m ≤ 4 is algebraic by exploring a finiteness condition.
Cecil, Thomas E.; Quo-Shin Chi and Gary Jensen. On Kuiper's question whether taut submanifolds are algebraic. Pacific J. Math. 234 (2008), no. 2, 229-248.