Let $F=\alpha + \beta$ be the Randers metric on manifold $M$ of dimention $n$. Here $\alpha$ is a Riemannian metric on $M$ and $\beta$ is a 1-form on $M$. Let $V=V^i \frac{\partial}{\partial x^i}$ be a vector field on $M$.
My Question: Is horizontal covariant differentiation $V$ with respect to F the same the same covariant differentiation $V$ with respect to $\alpha$?