If $(X,d)$ is a metric space and $f : X \rightarrow \mathbb{R}$ is a Lipschitz function with Lipschitz constant $k < 1$, then the function $$ D(x,y) := d(x,y) + f(y) - f(x) $$ defines an asymmetric metric on $X$. I'll denote this procedure---for reasons I'll explain later---as adding a coboundary to $d$

Note that if $x,y$, and $z$ are three points in $X$ satisfying $d(x,y) + d(y,z) = d(x,z)$, then they also satisfy $D(x,y) + D(y,z) = D(x,z)$, which entails that the geodesics in $(X,d)$ are also geodesics in $(X,D)$.

Question. Are there interesting examples of finite or infinite graphs $\Gamma$ (perhaps some Cayley graph) such that if $D$ is an asymmetric metric with the same geodesics as the standard metric on $\Gamma$, then $D$ is obtained by adding a coboundary to a multiple of the standard metric?

Motivation for the problem and its title

I am looking for a wider setting and/or interpretation of a theorem of mine that generalizes what Gautier Berck and I did for projective Finsler metrics:

Theorem. If $D$ is a continuous asymmetric metric on the $n$-sphere $S^n$ for which all great circles are geodesics, there exists a (continuous) function $f : S^n \rightarrow \mathbb{R}$ such that $$ D(x,y) - D(y,x) = f(y) - f(x). $$

In other words, $D$ is obtained from the (symmetric) metric $d(x,y) := (D(x,y) + D(y,x))/2$ (for which all great circles are also geodesics) by adding a coboundary. This settles Hilbert's fourth problem for continuous asymmetric metrics in $n$-dimensional projective space by reducing the problem to continuous symmetric metrics treated by Busemann, Pogorelov, and Szabo.

Reading an old paper of Kolmogorov (Skew-symmetric forms and topological invariants in Vol.1. of his collected works), which is basically Kolmogorov's early version of the Alexander-Spanier cohomology, suggested the following setup:

Given an asymmetric metric $D$, consider the anti-symmetric function $g(x,y) := D(x,y) - D(y,x)$ as a 2-cochain. If $D$ has the same geodesics as its symmetrization, $d(x,y) := D(x,y) + D(y,x)$, then the differential of $g$, defined as $$ \delta g \, (x,y,z) = g(y,z) - g(x,z) + g(x,y) \, , $$ vanishes on all triples $(x,y,z)$ in which $y$ is between $x$ and $z$ (i.e., $d(x,y) + d(y,z) = d(x,z)$). I guess one could say somewhat loosely that "the differential of $g$ vanishes on geodesics". In any case, this is exactly the condition for $D$ and its symmetrization to share the same geodesics

The very loosely stated question would then be

Under what general conditions can we guarantee that if the 2-cochain $g(x,y) := D(x,y) - D(y,x)$ vanishes on all geodesics of the symmetric metric $D(x,y) + D(y,x)$, then $g$ is a cocycle or even a coboundary?

One can even forget about metrics, prescribe a system of "geodesics" on $X$ by giving a subset
$$ G \subset X \times X \times X $$ (i.e. $(x,y,z) \in G$ if $y$ is between $x$ and $z$) and ask for the condition that a 2-cochain that vanishes on $G$ be a cocycle or a coboundary.


1 Answer 1


This is a further comment on the question, which I add for the sake of completeness.

A simple remark that came out of a discussion with Jimmy Dillies:

Proposition. If $D$ is any (asymmetric) length metric on the real line, then $D(x,y)-D(y,x)$ is a coboundary.

Proof. First we show that $g(x,y) = D(x,y) - D(x,y)$ is a cocycle (i.e., $\delta g \equiv 0$). Indeed, if $x \leq y \leq z$, then $g(x,y,z) = 0$ because $D(x,y) + D(y,z) = D(x,z)$ and $D(z,y) + D(y,x) = D(z,x)$. However, the order of $x$, $y$, and $z$ is not important because $\delta g$ is alternating.

Setting $f(x) = g(0,x)$ and using the equation $\delta g(0,x,y) = 0$, we obtain $$ g(x,y) = f(y) - f(x) = \delta f(x,y). $$


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