On the Gauss–Bonnet–Chern formula for real Finsler vector bundles

In this paper, we give a Gauss–Bonnet–Chern formula for real Finsler vector bundles with respect to any metric-compatible connection. The key idea is to modify any given metric-compatible connection to be a new metric-compatible connection with some special properties. As a corollary, a Gauss–Bonnet–Chern formula for Finsler manifolds with respect to any metric-compatible connection is established.