Definition
Consider a domain $\Omega \subseteq \mathbb{R}^3$ and three differentiable maps $f_i:\Omega \rightarrow \mathbb{R}$, $i = 1, 2, 3$. If at every point $x \in \Omega$, $\nabla f_i(x) \cdot \nabla f_j(x) \ne = 0$ only if whenever $i=j$, i \ne j$, then the $f_i$ are triply orthogonal coordinates on $\Omega$.
Context
Dupin's theorem states that orthogonal coordinate surfaces (i.e., level sets of orthogonal coordinate functions) intersect along lines of principal curvature. A classic example is Monge's ellipsoid -- see Jorge Sotomayor, "Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in \mathbb{R}^3."
Question
Suppose we start with a compact, connected, orientable smooth surface $\mathcal{M}$ embedded in $\mathbb{R}^3$, and let $\Omega$ be a band of uniform thickness around $\mathcal{M}$ (i.e., a union of $\epsilon$-balls centered at each point on $\mathcal{M}$). \mathcal{M}$) small enough so that at each point $x \in \Omega$ there is a unique closest point on $\mathcal{M}$. Further, suppose $f_1:\Omega \rightarrow \mathbb{R}$ gives the signed distance to $\mathcal{M}$, so that $f^{-1}(0) = \mathcal{M}$.
Can we always find (nontrivial) functions $f_2, f_3$ such that $(f_1,f_2,f_3)$ are triply orthogonal coordinates?
Or less formally, can we always find orthogonal coordinates "around" a given surface?
