If $(M,g)$ is a Riemannian manifold, $\alpha \in \Omega^1 M$ a smooth $1$-form, $x \in M$ and $y$ in a normal coordinates neighbourhood $U$ of $x$, how should I proceed in differentiating $y \mapsto \alpha_y (\exp_y ^{-1} (x))$ with respect to $y$? (I am interested in the gradient and the Laplacian of this function.)
If differentiating were with respect to $x$, I would know what to do (perform the computation in normal coordinates around $x$), but how to deal with the argument being the base point of $\exp$, so that for each $y$ there exists a different set of normal coordinates?