Reference for a formula of exterior derivative Often I come across this formula in the literature. Can someone please give me a proof or a reference where the proof is given with all the details? Thanks.
On a Riemannian manifold for any form $\beta,$ if $e_i$ the local co-frame then
$d\beta=\sum\limits_i e_i\wedge \nabla_i\beta$, where $\nabla_i$ is the covariant derivative in the corresponding direction w.r.t. the levi-Civita connection.
 A: As with many calculations, it depends on what you know so as to where you can begin. If you are happy with the following identity (summation convention used throughout):
$$\nabla_{X_{a}}e^{c} \,=\, -\omega^{a}_{\phantom{a}b}(X_{a})\,e^{b}$$
in terms of the covariant derivative $\nabla$, dual frame $\{X_{a}\}$ satisfying $e^{a}(X_{b})=\delta^{a}_{b}$, and where $\{\omega^{a}_{\phantom{a}b}\}$ are the connection $1$-forms, then it is relatively straightforward to show your identity using the definition of the torsion $2$-forms:
$$T^{a}\,=\,de^{a} + \omega^{a}_{\phantom{a}b} \wedge e^{b}.$$
Since you are dealing with the Levi-Civita connection: $T^{a}=0$ (though you can obtain an identity similar to the one you are after with non-vanishing torsion), then from the first displayed equation:
$$ e^{a} \wedge \nabla_{X_{a}}e^{c} \,=\, -\omega^{c}_{\phantom{a}b}(X_{a})e^{a} \wedge e^{b} \,=\, -\omega^{c}_{\phantom{a}b} \wedge e^{b}$$
and so
$$ T^{c} \,=\, 0 \,=\, de^{c} + \omega^{c}_{\phantom{a}b} \wedge e^{b} \,=\, de^{c} - e^{a} \wedge \nabla_{X_{a}}e^{c} \,=\, (d - e^{a} \wedge \nabla_{X_{a}})e^{c}$$
yielding the identity you desire. The nice thing is this also holds in pseudo-Riemannian geometry. As for a reference to this, I think I went through this calculation in "An Introduction to Spinors and Geometry with Applications in Physics" by Benn & Tucker where, from memory, the calculation is done in more generality than you probably need (pseudo-Riemannian and with torsion).
