# Is “Cartan's magic formula” due to Élie or Henri?

The formula $\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. Berline, Getzler, Vergne; Heat Kernels and Dirac Operators, p.17), and often just to "Cartan" (e.g. http://en.wikipedia.org/wiki/Lie_derivative ). Who is right? Reference?

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www22.pair.com/csdc/ed3/ed3fre8.htm gives a reference that suggests it's the senior Cartan. –  Qiaochu Yuan Sep 21 '10 at 21:26
It is almost certainly Élie Cartan. –  Deane Yang Sep 21 '10 at 21:34

Élie for sure. The formula is derived in "Les systèmes differentiels extérieurs et leur applications géométriques" which was probably written before Henri was born. BTW, here is a very short proof that Chern showed me long ago. The exterior derivative is an anti-derivation of the exterior algebra and so is the interior product with a vector field while the Lie derivative is a derivation. (These are all trivial to check.) Also, the anti-commutator of a derivation and a derivation is an anti-derivation. Hence both sides of the "magic formula" are anti-derivations. It is obvious that two anti-derivations are equal if they agree on 0-forms and 1-forms, since the latter generate the exterior algebra. Finally it is trivial that both sides of the magic formula agree on forms of degree 0 and 1.

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I'm not sure I agree that the formula is trivial for $1$-forms. To me that's the first (and only?) nontrivial case, and Chern's argument is a nice way to explain how to extend it to higher degree forms. But I'm just quibbling. –  Deane Yang Sep 21 '10 at 22:02
Sorry, Deane, you are correct---I screwed up. I should have said that it is obvious for functions $f$ and the differentials of functions, $df$, and that THESE generate the exterior algebra. The reason it is clear for $\omega = df$ is that the RHS becomes $di_X (df) = d(Xf)$. On the other hand, since $d$ is "natural", i.e., commutes with diffeos, it also commutes with Lie derivatives, so the LHS becomes $d L_X f)$ which is also $d(Xf)$. –  Dick Palais Sep 21 '10 at 22:23
/Les systèmes différentiels extérieurs et leurs applications géométriques/ seems to have been written in 1945. Henri was certainly born at this date. –  Maxime Bourrigan Sep 21 '10 at 22:27
I am THE Palais father. :-) BTW, do you know about our co-authored book? See: ode-math.com. –  Dick Palais Sep 21 '10 at 23:10
This formula already appears in É. Cartan's 1922 classic, Leçons sur les invariant intégraux. I believe it's in Chapter IX, where he discusses the effect of infinitesimal transformations' (i.e., vector fields) on differential forms. Of course, Henri was alive then, but he was only 18. –  Robert Bryant May 13 '11 at 22:30

It is due to Henri Cartan according to 3), Page 193 of: S.S. Chern, et al, Lectures on Differential Geometry, World Sci, Singapore, 2000.

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I think the earlier answers have already conclusively ruled this out. –  Deane Yang Mar 26 '12 at 13:47
Dear Deane, to be precise, I think that the final version is due to Henri. It is originally due to Elie while the name "Lie derivative" is suggested by K. Yano. –  Feng-Wen An Mar 26 '12 at 16:33
Could you say more about how the final version is different from the earlier versions? And where did Yano first use the term "Lie derivative"? –  Deane Yang Mar 26 '12 at 19:48
According to an article by Andrzej Trautman (2008), the general Lie derivative on tensors got introduced by Władysław Ślebodziński (1931) and named "Lie derivative" (in German) by David van Dantzig (1932). Trautman article: fuw.edu.pl/~amt/4Krupka.pdf –  Martin Gisser Mar 27 '12 at 22:53