I'd appreciate it a lot of you could please give me a detailed answer to this question. Alternately, you could just cite a reference too. Let $(M,g)$ be a Riemannian manifold, $c$ a curve on it. You could assume that $c$ is a geodesic if you need to, but I think you might not need it. Let $P_{t_1,t_2}$ denote the parallel transport from $P_{t_1,t_2}:T_{c(t_1)}M\to T_{c(t_2)}M$. Let $\nabla$ denote the covariant derivative for $(M,g)$ and let $R$ denote the $(3,1)$ Riemannian curvature tensor. Let $X,Y,Z$ be vector fields along $c$. Then are the following true:
1) Is $P_{t_1,t_2}(\nabla_{c'}Y(t_1))=\nabla_{c'}(P_{t_1,t_2}(Y(t_1))$ ?
2) Is $P_{t_1,t_2}(R(X,Y)Z)(t_1))=R( P_{t_1,t_2}(X(t_1)), P_{t_1,t_2}(Y(t_1))P_{t_1,t_2}(Z(t_1)) ?$
I think yes to 1) would imply yes to 2).
Please give a detailed answer or a reference/resource.Thanks!