I just started to read the book, and already like his explanation style. Thank you for the answer, it answers some other questions I had too.

I'm not familiar with metric compatibility on bundles. Is what you said translate to [X,Y] <s_i,s_j> = <R(X,Y)s_i, s_j> + <s_i, R(X,Y)s_j> = 0? Here s_i, s_j are smooth sections and X,Y are tangent vectors. R is the curvature.