Parallel transport

A vector field exists in a particular tangent space of a curved manifold. Parallel transport allows us to move a vector $V$ from one point on the manifold to another $A \to B$ through a series of infinitesimal steps, ensuring that the $V(A)$ and $V(A \to B)$ remain parallel.

The vector defined at $B$ with the same components $V^\alpha(B) = V^\alpha(A)$ will generally not be parallel to $V(A)$ except on a flat manifold.

We define the components of the parallel transport of $V$ in terms of the Christoffel connection.

\begin{align*} V^\alpha(A \to B) = V^\alpha(A) - \Gamma^\alpha_{\beta\gamma} dx^\beta V^\gamma. \end{align*}