# Transnormal function and isoparametric function

Let M be a connected complete Riemannian manifold and denote by $$\nabla$$ and $$\triangle$$ the Levi-Civita connection and the Laplace operator of M, respectively. A non-constant function f of class $$C^{2}$$ on $$M$$ is called transnormal if $$$$\label{RK 1} |\nabla f|^{2}=b(f)$$$$ for some real function b of class $$C^{2}$$ defined on the range of $$f$$, and $$f$$ is called isoparametric if, in addition to the first equation, it satisfies $$$$\label{RK 2} \triangle(f) = a(f)$$$$ for some continuous real function a defined on the range of $$f$$. The first equation implies that the level sets of $$f$$ are parallel to each other. The second equation implies that a level hypersurface has constant mean curvature. Qustion: If $$f$$ is transnormal function and the level sets of $$f$$ are parallel to each other, how to prove that $$f$$ is isoparametric function?

• It displays nicer if you use \Delta instead of \triangle for the Laplacian operator. – Alex M. Mar 25 '19 at 14:22