I am trying to show (Schutz chpt. 6 prob 13) that if two vector fields $\vec{A}$ and $\vec{B}$ are parallel transported along a curve $\gamma:\mathbb{R}\to M$ with real parameter $\lambda$ ($M$ a Riemannian manifold with metric $g$) then $g(\vec{A},\vec{B})$ is constant along the curve, $\gamma$.
So I want to show that $\frac{d}{d\lambda}g(\vec{A},\vec{B})=0$. Using the product rule this gives me: $$ \frac{d}{d\lambda}[g_{\alpha\beta}A^\alpha B^\beta]=\frac{d}{d\lambda}[g_{\alpha\beta}]A^\alpha B^\beta+g_{\alpha\beta}\frac{d}{d\lambda}[A^\alpha]B^\beta+g_{\alpha\beta}A^\alpha\frac{d}{d\lambda}[B^\beta] $$ This is the point where my confusion arises. If I denote the tangent vector to $\gamma$ by $\vec{U}$, is, for example, $\frac{d}{d\lambda}g_{\alpha\beta}=U^\mu g_{\alpha\beta,\mu}$ or $\frac{d}{d\lambda}g_{\alpha\beta}=U^\mu g_{\alpha\beta;\mu}$? It seems that in Schutz he writes that $\frac{d}{d\lambda}V^\alpha=U^\beta {V^\alpha}_{,\beta}$ (equation 6.47 pg. 156) and he says that this is "the definition of a derivative of a function along a curve", which seems to be the situation I am dealing with right now.
But if this is the case then I can't use the parallel transport condition $\nabla_{\vec{U}}V^\alpha=U^\beta {V^\alpha}_{;\beta}=0$ or the fact that $g_{\alpha\beta;\mu}=0$ which is what I would think that I should use. Also I would think that I should take a covariant derivative because this result should be independent of the coordinates I choose.
I guess I'm generally confused about differentiating along a curve- if anyone has Schutz then clearing up equation 6.48 would help me a bunch- he writes: the definition of the parallel transport of $\vec{V}$ along $\vec{U}$ is: $$ U^\beta {V^\alpha}_{;\beta}=0\iff\frac{d}{d\lambda}\vec{V}=\nabla_{\vec{U}}\vec{V}=0 $$ " (where $\vec{U}$ is the tangent vector to the curve in discussion). But if, like he says, $\frac{d}{d\lambda}\vec{V}=U^\beta {V^\alpha}_{,\beta}$, then I don't understand how we could ever have $\frac{d}{d\lambda}\vec{V}=\nabla_{\vec{U}}\vec{V}=0$ unless we are in a locally flat point or the Christoffel symbols are zero everywhere.
Any help would be GREATLY appreciated.