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. https://en.wikipedia.org/wiki/Lie_derivative ). Who is right? Reference?
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asked Sep 21, 2010 at 21:21
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3$\begingroup$ www22.pair.com/csdc/ed3/ed3fre8.htm gives a reference that suggests it's the senior Cartan. $\endgroup$– Qiaochu YuanCommented Sep 21, 2010 at 21:26
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4$\begingroup$ It is almost certainly Élie Cartan. $\endgroup$– Deane YangCommented Sep 21, 2010 at 21:34
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4 Answers
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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 two anti-derivations is a derivation. Hence both sides of the "magic formula" are derivations.
- It is obvious that two derivations are equal if they agree on 0-forms $f$ and exact 1-forms $df$, since (locally) they generate the exterior algebra.
- Finally it is trivial that both sides of the magic formula agree on such forms.
answered Sep 21, 2010 at 21:34
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4$\begingroup$ 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. $\endgroup$ Commented Sep 21, 2010 at 22:02
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15$\begingroup$ 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 $d(Xf)$. $\endgroup$ Commented Sep 21, 2010 at 22:23
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4$\begingroup$ /Les systèmes différentiels extérieurs et leurs applications géométriques/ seems to have been written in 1945. Henri was certainly born at this date. $\endgroup$ Commented Sep 21, 2010 at 22:27
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7$\begingroup$ I am THE Palais father. :-) BTW, do you know about our co-authored book? See: ode-math.com. $\endgroup$ Commented Sep 21, 2010 at 23:10
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7$\begingroup$ 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. $\endgroup$ Commented May 13, 2011 at 22:30
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Certainly Henri Cartan was too young to contribute to this formula and his father Elie played a decisive role, but it is not clear to decide who invented it. Indeed the formula can also be found in the Théophile De Donder's book "Théorie des invariants intégraux" published in 1927. Exterior differential forms are called there "formes intégrales", the exterior differential operator is called "différentielle intégrale" and is denoted by D. Formula DD=0 and Stokes' formula are setted. The main reference for that is H. Poincaré (with contributions by Volterra, Cartan, Goursat, De Donder). The exterior product is called "produit intégral" and is denoted by [AB] (as in E. Cartan's text). The interior product is called "substitution intégrale" and is denoted by E. De Donder indicates there that this notion was introduced by H. Poincaré. Then the magic formula is attributed to Edouard Goursat ( E. Goursat, Sur quelques points de la théorie des invariants intégraux, J. Math. Pures Appl. (7), t. I (1915), 241—259; Sur certains systèmes d'équations aux différentielles totales et sur une généralisation du problème de Pfaff, Ann. Fac. Sci. Toulouse, t. VII (1915)) and De Donder (Th. De Donder, Sur les invariants intégraux relatifs et leurs applications à la physique mathématique, Bull. Acad. Roy. Belgique, Classe des Sciences, fév. 1911, 50--70.).
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1$\begingroup$ Thanks for the history and references and welcome to MathOverflow! $\endgroup$ Commented Mar 26, 2012 at 10:28
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4$\begingroup$ I'll repeat my earlier comment to Dick's answer: "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." This is probably why it is called "Cartan's formula". I note that, in that work, Cartan attributes the definition of the exterior derivative itself to Goursat, but he doesn't attribute the above formula to anyone. $\endgroup$ Commented Mar 26, 2012 at 12:04
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1$\begingroup$ Dear @Robert: I suggest that you post an answer. $\endgroup$ Commented Mar 26, 2012 at 13:11
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1$\begingroup$ @Pierre: It looks as though Frédéric has found references that go back before Cartan's first mention of the formula in print (which appears to be 1922). I haven't gone to look at the Goursat and De Donder references that he cites, but, it appears that calling it 'Cartan's formula' (which, of course, Cartan never did) is historically inaccurate. My comment was mainly to speculate that the 1922 reference may be why we call it the Cartan formula today. It's odd that Cartan didn't credit Goursat (or De Donder?) in his 1922 book, since he certainly held him in great esteem. $\endgroup$ Commented Mar 26, 2012 at 15:34
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1$\begingroup$ @Martin: One reason that Trautman says `around 1920' is that, according to its preface, Cartan's 1922 book is the write-up of a course he gave in the 'summer semester' of 1920-1. Thus, presumably, he knew the formula by then. Note also that the equation (5) you are referring to in Cartan's book is, literally for a 3-form only, but it's clear that Cartan meant this to be an illustrative example of the formula for all degrees $p$. He didn't give the general derivation because his notation would have made it cumbersome, but he clearly knew that, in principle, the proof was the same for all $p$. $\endgroup$ Commented Apr 1, 2012 at 16:54
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Actually it seems that this formula was proved by Théophile De Donder in 1911, as you can see in his paper in Bulletins de la Classe des Sciences de l'Académie Royale de Belgique : https://www.biodiversitylibrary.org/item/273691#page/11/mode/1up , pages 50 to 70. The interior product is denoted by E there.
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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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6$\begingroup$ I think the earlier answers have already conclusively ruled this out. $\endgroup$ Commented Mar 26, 2012 at 13:47
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$\begingroup$ 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. $\endgroup$ Commented Mar 26, 2012 at 16:33
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$\begingroup$ 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"? $\endgroup$ Commented Mar 26, 2012 at 19:48
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3$\begingroup$ 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 $\endgroup$ Commented Mar 27, 2012 at 22:53