Fuction whose gradient is of constant norm on its level sets I have a function $f:\mathbb{R}^N \to \mathbb{R}$ and I know that on its level sets $f^{-1}(z)$ the norm of its gradient is constant. What can I say about this function?
$$
||\nabla_x f(x)|| = \text{const} \qquad \qquad \forall x \in f^{-1}(z) := \left\{x \in \mathbb{R}^N \, :\, f(x) = z\right\} \qquad \forall \in \mathbb{R}
$$
Related questions are this and this. However, they consider the norm of the gradient to be constant for every $x$ in the domain. I know that this is true only on each level set.
 A: If $f$ is $C^{1}$, then because the norm of the gradient is constant on levels of $f$, there is a continuous, real-valued function $\lambda$ of one variable satisfying
$$
\|\nabla f(x)\| = \lambda\bigl(f(x)\bigr)\qquad\text{for all $x$.}
$$
If $f$ has no critical points, then $\lambda > 0$. Let $\Lambda$ be an antiderivative for $1/\lambda$ and let $g = \Lambda \circ f$. By the chain rule,
$$
\|(\nabla g)(x)\| = \bigl|\Lambda'\bigl(f(x)\bigr)\bigr| \cdot \|\nabla f(x)\| = 1.
$$
By the solution of the linked question, $g$ is affine. Consequently, $f$ is constant on hyperplanes, and is therefore effectively a function of one variable.
If $f$ has critical points, more can happen. For example, $f$ could be a function of distance-squared from an affine subspace (a point up through a hyperplane).
[Musings: Offhand I don't have a proof that's all, but along the lines of Chris' comment this is what one expects; I'd be inclined to check whether the regular levels of $f$ have constant principal curvatures.]
