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In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for areal oriented Finsler vector bundle with rank equal to the dimension of thebase manifold. As an application, a Gauss-Bonnet-Chern formula for anymetric-compatible connection is established on Finsler manifolds.

A Gauss-Bonnet-Chern theorem for Finsler vector bundles