TITLE: Formula relating covariant derivative and exterior derivative
QUESTION [5 upvotes]: According to Gallot-Hulin-Lafontaine one has
$$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_1,\cdots,X_{i-1},X_0,X_{i+1},\cdots,X_q)$$
It seems to me that it should be
$$d\alpha (X_0,\cdots,X_q) = \sum_{i=0}^q (-1)^i D_{X_i} \alpha (X_0,\cdots,\hat{X_i},\cdots,X_q)$$
Is this right ?

REPLY [4 votes]: If $\theta$ is a 1-form, then
$$
d\theta(X,Y) = (\nabla_X\theta)(Y) - (\nabla_Y\theta)(X)
$$
If $\Omega$ is a 2-form, then
$$
d\Omega(X,Y,Z) = (\nabla_X\Omega)(Y,Z)+(\nabla_Z\Omega)(X,Y)+ (\nabla_Y\Omega)(Z,X)
$$
and so on ... but you have to have a zero-torsion (symmetric) connection. These formulae will be useful when manipulating structure equations, for instance to obtain Bianchi identities. -- Salem