I am examining the covariant derivative of a vector according to the formula $$\nabla_{\mu}V^{\nu} = \partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$ and also operating under the assumption that the covariant derivative is a map between $(k,l)$ and $(k,l+1)$ tensors. Now, I have seen that the definition for parallel transport is:
a vector $v^a$ is parallely transported along a curve if the following equation is satisfied along the curve
$$t^a\nabla_av^b = 0$$
However, $\nabla_av^b = T^b_a \rightarrow t^a\nabla_av^b = V^b$. Thus, the result of this parallel transport equation should be a vector. Conceptually, then, our "restriction" is that $t^aT^b_a = 0$. I am not sure how to visualize this. Carroll claims that this restriction can be seen as saying the covariant derivative of the vector vanishes along the curve. However, this restriction does not seem to demonstrate that to me. I am wondering how to interpret this equation based on the mathematical formulation of it.
