Let M be a Riemmanian manifold. Prove that a parallel vector field along a curve c(t) preserves the length of the parallel transported vector. Furthermore if M is an oriented manifold, prove that P preserves the orientation. So I want to prove that d/dt <P, P> = 0, so <P, P> = constant. d/dt <P, P> = 2<dP/dt, P> But dP/dt is not an element of the tangent space at c(t) for any t. DP/dt is however always an element of the tangent space, but since DP/dt is 0, we get our desired result. But am I allowed to do say that dP/dt is in general not an element of the tangent space? DP/dt is always an element of the tangent space as we can see when we use a local expression for it. However, I feel like I am using an exterior argument by essentially saying that: dP/dt = orth(dP/dt) + tang(dP/dt) i.e. I'm splitting it up into tangential and orthogonal components. Then when we "dot" an orthogonal part with the purely tangential part, it drops out and we are left with: <DP/dt, P> = 0. Is there another way to do this in terms of local coordinates? And if M is an oriented manifold, what does it mean exactly for P to preserve orientation? Thanks guys!