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In typical studies of Finsler geometry, the metric function $F: TM \to [0,\infty)$ is assumed to be smooth away from the zero section, and $F$ is assume to be sufficiently convex. Under these assumptions the geodesic spray is well-defined.

  1. Formally writing down the Euler-Lagrange equation for the length minimizer one sees that to apply Picard-Lindelof we would want $F$ to be at least $C^{2,1}$ (for simplicity let's work on a local coordinate patch).
  2. The convexity comes in because the ELE looks formally like $$ A(\gamma, \dot{\gamma}) \cdot \ddot{\gamma} = N(\gamma, \dot{\gamma}) $$ where $\gamma$ is vector valued and $A$ is a matrix of coefficients; the convexity would ensure that $A$ is invertible.

Are there any articles/surveys that describes (preferably with examples), what exactly goes wrong to the geodesic structure of the manifold if either $F$ is too rough, or if $F$ has flat points? Naively, I would expect to see something like the following:

  1. If $F$ is too rough, there may be some sort for "refraction" in the geodesic spray.
  2. If $F$ is insufficiently convex (so that $A$ is singular), locally the geometry picks up some sub-Riemannian flavor.

Has this been examined in the literature?

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  • $\begingroup$ I'd love to know why you're interested in this. I think it's instructive to start by looking at the $\ell_1$ and $\ell_\infty$ norms on $\mathbb{R}^n$. In the latter case, you get the so-called taxi metric, so there are infinitely many piecewise linear geodesics and limits that are non-differentiable. I think you're right about a sub-Riemannian flavor, mostly due to the flat parts of the unit ball. $\endgroup$
    – Deane Yang
    Feb 10, 2021 at 17:12
  • $\begingroup$ I believe the most important Finsler geometric invariants are not differential, as for Riemannian geometry. The deepest invariants of an arbitrary convex body arise from integral geometry. To be a Finsher invariant, these need to be affine invariant (i.e., invariant under change of basis in the tangent space). There are many such integral variants, and when you look at their variational formulas, you get curvature measures on a tangent bundle. I like to call these invariants "micro-integral" geometric, as a play on the term "microlocal". $\endgroup$
    – Deane Yang
    Feb 10, 2021 at 17:16
  • $\begingroup$ @DeaneYang: for the first comment it actually grew out of a Twitter discussion, believe it or not. See qnlw.info/post/finsler-rod-balls-2021-02; the "metric" is the sum of the distance of two sub Riemannian metrics and is positive definite, but only $C^{0,1}$ and has some "flat points". After thinking about it for a couple of days I became interested in the more general cases. $\endgroup$ Feb 10, 2021 at 19:27
  • $\begingroup$ @DeaneYang: for the second comment... your lost me starting the second sentence. If you have a good reference that explains those ideas I'd appreciate it. $\endgroup$ Feb 10, 2021 at 19:28
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    $\begingroup$ My second comment was indeed cryptic. The idea is the following: If you have a Finsler manifold, then any quantity defined in terms of the unit tangent ball that is invariant under linear transformations defines a pointwise geometric invariant of the Finsler metric. An example is the Holmes-Thompson volume form. Once you have fixed a volume form on a vector space, then there are non-trivial linearly invariant integral invariants of a convex body, which themselves arise from homogeneous measures. Part of the story is told here: math.nyu.edu/~yangd/papers/affine_survey.pdf $\endgroup$
    – Deane Yang
    Feb 11, 2021 at 20:47

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I think a starting point could be

Minguzzi, E., Convex neighborhoods for Lipschitz connections and sprays, Monatsh. Math. 177, No. 4, 569-625 (2015). ZBL1330.53021.

He does a detailed analysis of the Picard-Lindelöf argument and it works for any signature.

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  • $\begingroup$ Thanks, but the situation I am thinking about is precisely when the connections fail to be Lipschitz. The function $F$ being in $C^{2,1}$ implies that both the matrix $A$ and the lower order terms $N$ in the Euler-Lagrange equations are Lipschitz in their arguments. $\endgroup$ Feb 10, 2021 at 16:39
  • $\begingroup$ And in terms of signature: Minguzzi's notion of a pseudo-Finslerian manifold requires the fundamental tensor to be non-degenerate, which is analogous to the convexity condition that I alluded to; it in particular implies that the matrix $A$ in my post is invertible. $\endgroup$ Feb 10, 2021 at 16:46
  • $\begingroup$ Yes, sorry. I thought you were also interested in going from smooth to $C^{2,1}$. $\endgroup$ Feb 10, 2021 at 16:50

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