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I am looking for a inner metric on $\mathbb{R}^2$ (that induces the standard topology) which is not Finsler.

By "Finsler" here I mean a metric that is obtained by the following construction:

  1. pick a smooth structure on $\mathbb R^2$ and take a suitable continuous function $\mu$ on $T\mathbb R^2$
  2. define a metric as $d(x,y):=\inf_\gamma \int \mu(\dot\gamma) dt $ over all piecewise smooth paths $\gamma$ connecting two points $x,y$.

If exists, it is necessarily non-invariant (Berestovski theorem).

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    $\begingroup$ What about the geodesic metric on the cone $x^2+y^2=z^2$, $z\ge 0$ (which is homeomorphic to the plane?) Or similarly, the quotient of $\mathbf{R}^2$ by $x\mapsto -x$, with distance induced by $\min(d(x,y),d(x,-y))$. $\endgroup$
    – YCor
    Commented Nov 29, 2020 at 19:25
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    $\begingroup$ The real question (originally due to Busemann) is "what is the characterization of Finsler distance functions among path-metrics on surfaces?" See "Foundations of singular Finsler geometry" by P.Andreev for some recent progress in this direction. $\endgroup$
    – Moishe Kohan
    Commented Nov 29, 2020 at 22:31
  • $\begingroup$ @MoisheKohan it's somewhat open-ended. Characterization in which terms? $\endgroup$
    – YCor
    Commented Nov 29, 2020 at 22:59
  • $\begingroup$ @YCor: Similar to Nikolaev's theorem characterizing Riemannian metrics among path metrics. The first condition would be local extendibility of geodesics, which fails in both examples that you gave. Compare my answers here and here. $\endgroup$
    – Moishe Kohan
    Commented Nov 29, 2020 at 23:33
  • $\begingroup$ See also 2nd answer here. $\endgroup$
    – Moishe Kohan
    Commented Nov 30, 2020 at 1:32

1 Answer 1

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Your metric has to be geodesic, in particular $$d(x,y)=\min \{\,1,|x-y|\,\}$$ is not Finsler in your sense.

Now let $d$ be a geodesic metric on $\mathbb{R}^n$.

Suppose $n=2$. In "Two counterexamples..." by Burago, Ivanov, and Shoenthal, it was conjectured that a neighborhood of any point in $(\mathbb{R}^2,d)$ admits a Lipschitz embedding into the Euclidean plane.

Suppose $d$ is a Finsler metric in your sense. Since $\mu$ is continuous, the natural map $(\mathbb{R}^n,d)\to \mathbb{R}^n$ is a locally lipschitz homeomorphism. So this conjecture is closely related to your question.

Now suppose $n=3$. The same paper provides an example of a metric $d$ on $\mathbb{R}^3$ that (locally) does not admit a Lipschitz embedding into $\mathbb{R}^3$. In particular, $d$ is not a Finsler metric in your sense. (The construction is interesting --- it is worth reading.)

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