If $(M,F)$ is an $n$-dimensional Finsler manifold, $\{ e_i \} $ a local orthonormal frame for $\pi^*TM$, $\{ \omega^i \}$ the dual coframe, and $$ \begin{align} C_{ijk}(x,y) &:= \frac{1}{4}e_k(e_j(e_i(F^2(x,y)))), \\ g_{ij} (x,y) &:= \frac{1}{2} e_j (e_i (F^2 (x,y))), \\ \omega^{n+i}&:= dy^i + y^j \omega^i_j, \ \text{where } y^ie_i \text{ is the canonical vector field on } \pi^*TM,\end{align} $$ then the Chern connection $\nabla$ is the unique linear connection on the pullback bundle with the following form $\omega^i_j$ (defined by $\omega^i_j(v)e_i:=\nabla_v e_j$) satisfying

$$ \begin{align} d\omega^i &= \omega^j \wedge \omega^i_j, \\ dg_{ij} &= g_{kj} \omega^k_i + g_{ik} \omega^k_j + 2C_{ijk} \omega^{n+k}. \end{align} $$

The first equation expresses 'torsion freeness'. I am guessing the second equation is to do with compatibility with the norm. Can anyone provide me with a complete geometric description for this (sources will do), analagous to the Riemannian case?


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