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We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$as a linear connection $\nabla^V$ on any open subset $\Omega\subset M$associated to any vector field $V$ on $\Omega$ which is non-zero everywhere.This connection is torsion-free and almost metric compatible with respect tothe fundamental tensor $g$. Then we show some properties of the curvaturetensor $R^V$ associated to $\nabla^V$ and in particular we prove that theJacobi operator of $R^V$ along a geodesic coincides with the one given by theChern curvature.

Chern connection of a pseudo-Finsler metric as a family of affine connections