Let $F$ be a Finsler metric and $g$ a Riemannian metric for $M$. Is there on Finsler manifolds a similar curvature to the mean curvature of Riemannian manifolds, such that if $F=\sqrt{g}$ then both curvatures are the same?
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$\begingroup$ What books are those definitions shows equivalence with the definition for riemannian manifolds? $\endgroup$– Otaner EncoCommented Sep 6, 2016 at 23:12
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I found a definition of the mean curvature of a hypersurface in Finsler geometry by Z. Shen in his book: Lectures on Finsler geometry. You will find it in chapter 14. To be honest I did not understand it till the moment yet. If you read it and have any comments, please let me know.
The definition is quite clever. Instead of worrying about "normal directions" which definition is not as clear-cut in the Finsler case, Shen use as a proxy a foliation based on a "distance function". Basically, given a hypersurface $N \subset M$ where $M$ has a Finsler structure $F$, Shen defines the mean curvature as follows:
- First find a (local) solution $\rho \in C^\infty(M)$ to the Eikonal equation $F(\nabla \rho) = 0$, when $\rho|_N = 0$. The gradient of $\rho$ then should correspond to some sort of normal direction to $N$.
- The Finsler structure on $M$ induces a notion of area on the level sets of $\rho$, which are hypersurfaces.
- The mean curvature is then the "log-derivative" of the area, analogously to how it is defined via area variation in Riemannian geometry.
