How to prove that any vector bundle with a fiber metric g admits a metric connection? It seems I should use partition of unity, but I have no idea how to proceed.
Also it seems there are two definitions of connection on a vector bundle E, one is that $\nabla: \Gamma(TM) \times \Gamma(E) \to \Gamma$(E), and the other one is that $\nabla: \Gamma(E) \to \Gamma(E) \otimes\Gamma(T^\ast M)$, and they should be equivalent. But I don't see how to use this to show the following equations are equivalent:
If there is a fiber metric g on vector bundle E, then
$d(g(u,v))=g(\nabla u,v)+g(u,\nabla v)$ for all $u,v\in\Gamma(E)$
is equivalent to
$X(g(u,v))=g(\nabla_X u,v)+g(u,\nabla_X v)$ for all $X\in TM$,
is it because that $d(g(u,v))(X)=X(g(u,v))$? I am just not sure about how vector fields act on metrics.