Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, and consider a one-form $\alpha\in\Omega^1(M;\mathbb{R})$ and a vector field $X\in\Gamma(TM)$.
Since $\alpha(X)\in C^\infty(M;\mathbb{R})$ we can consider it's differential $d(\alpha(X))$. Is there a coordinate invariant way to computer this quantity? Maybe something like $d(\alpha(X))(Y) = d\alpha(X,Y) + \alpha(\nabla_{Y}X)$? How does this generalize if we consider the differential of the smooth function $\beta(X,Y)$ where $\beta$ is now a 2-form and $X,Y$ are both vector fields?
Best Answer
Since ${\rm d}(\alpha(X))(Y) = Y(\alpha(X))$, we indeed get from $${\rm d}\alpha(X,Y) = X(\alpha(Y)) - Y(\alpha(X)) - \alpha([X,Y])$$that $${\rm d}(\alpha(X))(Y) = X(\alpha(Y)) - \alpha([X,Y]) - {\rm d}\alpha(X,Y),$$but this is not very satisfying as we still have $X(\alpha(Y))$ on the right side. Since $\nabla$ is torsion free, we may improve the right side to obtain $${\rm d}(\alpha(X))(Y) = (\nabla_X\alpha)(Y) + \alpha(\nabla_YX) - {\rm d}\alpha(X,Y),$$where $(\nabla_X\alpha)(Y) = X(\alpha(Y)) - \alpha(\nabla_XY)$ is the covariant derivative of $\alpha$ in the direction of $X$. So if $\iota_X$ denotes interior product, we can write $$ {\rm d}(\alpha(X)) = \nabla_X\alpha + \alpha \circ \nabla X - \iota_X({\rm d}\alpha), $$which is as coordinate-free as we can get. I'm not sure how this would actually be useful in practice, though. One can do the same and get more complicated formulas for $k$-forms with $k>1$ by using the same strategy, by using that if $\omega$ is a $k$-form, then $\nabla_X\omega$ is a $k$-form with $$(\nabla_X\omega)(X_1,\ldots, X_k) = X(\omega(X_1,\ldots,X_k)) - \sum_{i=1}^k \omega(X_1,\ldots, \nabla_XX_i,\ldots, X_k).$$You will also have to use $$\begin{align} {\rm d}\omega(X_0,\ldots,X_k) &= \sum_{i=0}^k (-1)^k X_i(\omega(X_0,\ldots, \widehat{X_i},\ldots, X_k)) \\ &\qquad + \sum_{0\leq i<j\leq k} (-1)^{i+j} \omega([X_i,X_j],X_1,\ldots, \widehat{X_i},\ldots, \widehat{X_j},\ldots, X_k). \end{align}$$