# Symmetric version of eikonal equation for pairwise distance function?

Suppose $$\mathcal M\subseteq\mathbb R^n$$ is a submanifold of $$\mathbb R^n$$. Take $$d(\cdot,\cdot):\mathcal M\times\mathcal M\to\mathbb R_+$$ to be the pairwise geodesic distance function, so $$d(x,y)$$ is the length of the shortest path from $$x\in\mathcal M$$ to $$y\in\mathcal M$$.

Holding one coordinate fixed and varying the other, wherever $$d(\cdot,\cdot)$$ is differentiable it satisfies the eikonal equation $$\|\nabla f(x)\|_2=1$$. In particular, we can write two conditions satisfied by $$d(\cdot,\cdot)$$ almost everywhere: $$\|\nabla_x d(x,y)\|_2=1\qquad\textrm{and}\qquad\|\nabla_y d(x,y)\|_2=1.$$

In a sense, these two conditions are redundant. If $$d(\cdot,\cdot)$$ satisfies the first condition, it "looks" like a distance function from $$y$$ to all the other points $$x$$ and the other condition follows by symmetry of $$d$$.

Is there a single, more symmetric PDE satisfied by $$d(\cdot,\cdot)$$ as a function on the product manifold $$\mathcal M\times\mathcal M$$?

That is, if the eikonal equation is the PDE behind the single-source-all-destinations geodesic distance problem, is there a different canonical PDE that governs the pairwise geodesic distance problem? I'm hoping to identify a condition that doesn't require enforcing the eikonal condition in the $$x$$ and $$y$$ coordinates individually.

This is not rigorous, but it may help.

Consider the Euclidean case, where $$d(x,y) = |x-y|$$. Then

$$\nabla d = \frac{(x-y, y-x)}{|x-y|},$$

where $$\nabla$$ denotes the gradient on $$\mathbb{R}^{2n}$$. Hence,

$$|\nabla d|^2 = \frac{|x-y|^2 + |y-x|^2}{|x-y|^2} = 2.$$

So, a necessary condition is

$$|\nabla d|^2 = 2,$$

subject to $$d(x,x) = 0$$.

In the Riemannian case, we likewise have

$$\nabla d = (\nabla_x d, \nabla_y d),$$

where $$\nabla_x$$ and $$\nabla_y$$ denote the gradient with respect to the first and second arguments, respectively. Hence,

$$|\nabla d|^2 =|\nabla_x d|^2+|\nabla_y d|^2 = 1+1 = 2.$$

So then, up to a factor 2, the distance function must satisfy the usual eikonal equation on the product space, subject to Dirichlet boundary conditions

$$d(x,x) = 0.$$

Since this equation has a unique (viscosity) solution, it should characterize the distance function.

• Awesome! Exactly what I was looking for. You rock, as always! Apr 9, 2020 at 18:05