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There is a formula connecting the exterior derivative and the Lie bracket $$d\omega (X,Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]).$$

What is a good way to remember this? By which I mean, what structure does this reveal? (Or, what essentially is going on here?) In his book, John Lee says, "In a sense, the Lie bracket is dual to the exterior derivative." But that's not really satisfying to me.

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  • $\begingroup$ I'm still wondering in what sense the Lie bracket is dual to the exterior derivative. $\endgroup$ – Andrews Apr 17 at 13:54
  • $\begingroup$ You should not remember this because this is not the formula followed at all places. For example in Kobayashi and Nomizu, what you said does not hold but $d\omega(X,Y)=\frac{1}{2}(X(\omega(Y))-Y(\omega(X))-\omega([X,Y]))$.. This is because their wedge product is defined differently... $\endgroup$ – user537667 Aug 14 at 19:15
  • $\begingroup$ “In particular, it shows that if we know all the Lie brackets of basis vector fields in a smooth local frame, we can compute the exterior derivatives of the dual covector f ields, and vice versa.” Page 369, 2nd edition of Lee’s smooth manifold book $\endgroup$ – user537667 Aug 14 at 20:28
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Eric, if you take $X=\partial/\partial x^i$, $Y=\partial/\partial x^j$, and $\omega = f dx^k$, then of course the formula gives you what you expect from the first two terms, and the bracket term disappears because $[X,Y]=0$ in this case. Why is the bracket term there in general? It's because $d\omega$ is a tensor field, and therefore must be linear over the $C^\infty$ functions. However, $X(\omega(Y)) - Y(\omega(X))$ is not linear over the $C^\infty$ functions, and it is the bracket term that precisely corrects for that. In general, formulas such as this that one can guess by applying to coordinate vector fields have bracket terms appearing exactly for that reason.

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  • $\begingroup$ Sir, +1... Can you share your opinion on my answer... Thanks.. $\endgroup$ – user537667 Aug 14 at 19:20
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I cannot improve on Ted Shifrin's answer describing why the bracket term is necessary. So, I'll only attempt in this answer to elaborate the sense in which the exterior derivative and bracket are dual.

Fix a local frame $(E_a)$ and let $(\theta^a)$ denote its dual coframe, so that $\theta^a (E_b) = \delta^a{}_b$; in particular each such contraction is constant. Then, for frame and coframe elements the exterior derivative formula simplifies to $$\phantom{(ast)} \qquad d\theta^a(E_b, E_c) = -\theta^a([E_b, E_c]) . \qquad (\ast)$$ But the right-hand side is (up to sign) exactly the structure function $C^a{}_{bc}$ of the frame, and the structure functions tell us everything there is to know about the geometry of the frame $(E_a)$, or equivalently the coframe $(\theta^a)$. Put another way:

The exterior derivatives $d\theta^a$ contain the same information encoded by the Lie bracket but instead expressed in the (dual) language of the dual coframe.

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