Proving $d\omega(X,Y) = X\omega(Y)-Y\omega(X) - \omega([X,Y])$ without choosing local coordinates.

Question

I'm going over some past exercises in differential geometry, and I'm struggling with going from a choice of local coordinates to a coordinate-free mindset.

In this exercise, I am to prove for a $1$-form $\omega$ and arbitrary vector fields $X,Y$ that $$d\omega(X,Y) = X\omega(Y)-Y\omega(X)-\omega([X,Y]).$$

In order to do this, we may restrict to local coordinates and write $\omega = \omega_i dx^i$, $X = X^i\partial_i, Y = Y^i \partial_i$ to get on the LHS $$ d\omega(X,Y) = \partial_j \omega_i(X^j Y^i - Y^j X^i) $$ and on the RHS expand similarly $X\omega(Y), Y\omega(X)$ and $\omega([X,Y])$. Eventually, stuff cancels out and we see both sides are equal.

Now, I am asked to generalize this formula for arbitrary $p$-forms $\omega$ acting on $p$ vector fields $X_1,\dots,X_p$, if possible. My approach above seems both cumbersome and ugly (even for $p=1$), and even if it generalizes to $p$-forms, I would prefer not to do the whole 'choose local coordinates'-tango.

Could someone show me how to do prove the $p=1$ case in a way that generalizes better to arbitrary $p$-forms?

Answer 1

For $S$ a $q$-form and $X,X_1,\ldots, X_q$ vectors fields, you can show that : \begin{align} (\mathcal{L}_XS) (X_1,\ldots, X_q) = X\cdot S(X_1,\ldots,X_q) - \sum S(X_1,\ldots, [X,X_i],\ldots,X_q) \end{align} Then you can show by induction that for a $p$-form $\alpha$ : \begin{align} \mathrm{d}\alpha (X_0,\ldots,X_p) =& \sum_{i=0}^p (-1)^i X_i \cdot \alpha(X_0, \ldots, \hat{X_i},\ldots,X_p) \\ &+ \sum_{0 \leqslant i < j \leqslant p} (-a)^{i+j}\alpha([X_i,X_j], X_0,\ldots, \hat{X_i}, \ldots, \hat{X_j},\ldots, X_q ) \end{align} where $\hat{X}$ means that you cancel the variable $X$. You do not have to use coordinates : it is the definition for $p=0$, you show it easily for $p=1$ and using the first equality above you can do induction.

Answer 2

Section 20.6 of 'An introduction to Manifolds' (2nd edition) by Loring W. Tu is related to your answer, and contains some precise proofs. Even in this book, local coordinates are used. However, you can reduce the proof to the case $\omega = f dg$, for $f,g \in C^{\infty}(U)$ thanks to linearity. The more general formula can be proved by induction: $$ d \omega (X_0, ..., X_n) = \sum_{i=0}^{n} (-1)^i X_i \omega (X_0, ..., \widehat{X}_i, ..., X_n) + \sum_{0 \leq i < j \leq n} (-1)^{i+j} \omega([X_i,X_j],X_0,...,\widehat{X}_i,...,\widehat{X}_j,..., X_n) $$ Here, the hat indicates deletion.