Neither of the two most evident sources,

seems to provide me with the high-level intuition that I could convey to students in ~10 minutes. And that is because I myself am not sufficently schooled in Finsler geometry.

Either a pointer to a better source, our your own take, would be welcomed. Thanks!

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    $\begingroup$ Perhaps you would enjoy ams.org/notices/199609/chern.pdf $\endgroup$ – Steven Gubkin Sep 11 '14 at 1:15
  • $\begingroup$ @StevenGubkin: Perfect, Steven---Thanks! $\endgroup$ – Joseph O'Rourke Sep 11 '14 at 1:18
  • $\begingroup$ So, is the answer simply the title of the linked article. I think that comes in at about 10 seconds, well short of the 10 minute mark. $\endgroup$ – James S. Cook Sep 11 '14 at 5:00
  • $\begingroup$ Why do you want to teach either students or yourself about Finsler metrics? $\endgroup$ – user173 Sep 12 '14 at 4:33

The easiest examples of Finsler metrics are manifolds where the length of a curve is defined by

$$\int \sqrt[\large{4}]{\dot{x}^4 + \dot{y}^4} dt.$$

Distances are then defined by the corresponding infimum. With this in mind the definitions are easier.

  • $\begingroup$ Thanks, Matt, this is extremely helpful to have such a clear example! I wanted to contrast Riemannian metrics which are locally Euclidean with Finsler metrics which are not (necessarily). And to discuss geodesics on Finsler metrics. $\endgroup$ – Joseph O'Rourke Sep 12 '14 at 21:21

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