Isoparametric functions on Riemannian manifolds. I (Q1099421)

A smooth function \(f: M\to {\mathbb{R}}\) on a complete Riemannian manifold M is called transnormal if \(\| df\|^ 2=b(f)\), b smooth. The main result of this paper says that all level sets of a transnormal function are smooth submanifolds. This has the following surprising consequence: In E. Cartan's definition of an isoparametric function f on \(S^ n\) or \({\mathbb{R}}^ n\) \((\| df\|^ 2=b(f)\), \(\Delta f=c(f))\) one can drop the condition on \(\Delta\) f, provided that f is globally defined.