A isoparametric function $f$ on a Riemannian manifold is a function that satisfies the followings identities: $|\nabla f|=a(f)$ and $\Delta f=b(f)$. Now, I would like to deal with hypersurfaces in the Euclidean ambient.
In fact, let $\Sigma^n$ be a hypersurface immersed in $\mathbb{R}^{n+1}$ and $h:\Sigma\rightarrow\mathbb{R}$, defined by $h(p)=\langle p, e_{n+1}\rangle$, where $e_{n+1}$ is a unitary vector, the height function on $\Sigma.$
If the function $h$ is isoparametric on $\Sigma$, is it possible to obtain some information about $\Sigma$?