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It seems that this problem is studied under the heading of Lamé families, i.e., those families of surfaces belonging to an orthogonal triple. Cayley was the first to give a general characterization of Lamé families in terms of a system of 3rd-order differential equations -- a simplified form can be found on p. 466 of A. R. Forsyth, "Lectures on the Differential Geometry of Curves and Surfaces.". In his paper, "On Lamé Families of Surfaces" , (1926), C.E. Weatherburn gives an even simpler characterization via the following condition on the unit normal field $n$:

$$\mathrm{div}(\frac{\partial}{\partial n} \mathrm{curl} mathrm{curl}\ n) = 0.$$

The

In the paper, the author applies this condition as though if it is were sufficient to characterize a Lamé family (see, e.g., the paragraph on p. 303 following the statement of the theorem), but from the proof it seems like only a necessary condition. In any case, in the book, "Differential Geometry of Three Dimensions" the same author states,

"A necessary and sufficient condition that a family of surfaces may form part of a triply orthogonal system is that, on each surface, the vector $\bar{\nabla}\psi$ be the gradient of some scalar function." (p. 101)

Here $\psi = ||n||$, and the operator $\bar{\nabla}$ is given by $\bar{\nabla}\phi = -\nabla \phi \cdot \nabla n$ where $\nabla$ is the surface gradient operator. In particular (he notes), any parallel family of surfaces is a Lamé family. So the answer to the original question is positive (since the distance function $f_1$ defines a parallel family).

1

It seems that this problem is studied under the heading of Lamé families, i.e., those families of surfaces belonging to an orthogonal triple. Cayley was the first to give a general characterization of Lamé families in terms of a system of 3rd-order differential equations -- a simplified form can be found on p. 466 of A. R. Forsyth, "Lectures on the Differential Geometry of Curves and Surfaces." In his paper, "On Lamé Families of Surfaces", C.E. Weatherburn gives an even simpler characterization via the following condition on the unit normal field $n$:

$$\mathrm{div}(\frac{\partial}{\partial n} \mathrm{curl} n) = 0.$$

The author applies this condition as though it is sufficient to characterize a Lamé family (see, e.g., the paragraph on p. 303 following the statement of the theorem), but from the proof it seems like only a necessary condition. In any case, in the book, "Differential Geometry of Three Dimensions" the same author states,

"A necessary and sufficient condition that a family of surfaces may form part of a triply orthogonal system is that, on each surface, the vector $\bar{\nabla}\psi$ be the gradient of some scalar function." (p. 101)

Here $\psi = ||n||$, and the operator $\bar{\nabla}$ is given by $\bar{\nabla}\phi = -\nabla \phi \cdot \nabla n$ where $\nabla$ is the surface gradient operator. In particular (he notes), any parallel family of surfaces is a Lamé family. So the answer to the original question is positive (since the distance function $f_1$ defines a parallel family).