It is well known that the image of a free Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$ on a manifold $M$ is an integrable distribution of constant rank $(=\rm{dim}\;\mathfrak{g})$. Thus it defines a foliation $\mathcal{F}$ on $M$. The leaf of $\mathcal{F}$ are called the $\mathfrak{g}$-orbits of the action. Conversely, given a foliation $\mathcal{F}$ on a manifold $M$, what are the conditions on $\mathcal{F}$ to be the foliation of a (free) Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$, i.e., the leaves of $\mathcal{F}$ are the $\mathfrak{g}$-orbits of $\rho$ ?
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$\begingroup$ If you let $\mathfrak g$ be the Lie algebra of fields tangent to the foliation and $\rho$ the inclusion, you always have that the foliation is the set of $\mathfrak g$-orbits, no? $\endgroup$– Mariano Suárez-ÁlvarezCommented Jan 15, 2012 at 21:36
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$\begingroup$ (I guess, though, you want the action of $\mathfrak g$ to be free, as in the first sentence?) $\endgroup$– Mariano Suárez-ÁlvarezCommented Jan 15, 2012 at 21:51
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$\begingroup$ Yes, I want the action to be free, so $\rm{dim}\;\mathfrak{g}$=dimension of the foliation $<\infty$. The Lie algebra of fields tangent to the foliation is infinite-dimensional. $\endgroup$– ZouhairCommented Jan 15, 2012 at 22:55
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2$\begingroup$ Besides the obvious conditions that the subbundle $D\subset TM$ of vectors tangent to the leaves be trivial and integrable, I doubt that there is much you can say. If you want the action to be complete, each leaf of $\mathcal{F}$ will have to have the topology of a discrete quotient of a fixed Lie group $G$, and, for example, that will rule out leaves having the topology of the $7$-sphere, even though it is parallelizable. It's probably not even that easy to say when an integrable $2$-plane field $D\subset TM$ that is smoothly trivial can be spanned by two commuting vector fields $X$ and $Y$. $\endgroup$– Robert BryantCommented Jan 16, 2012 at 0:23
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1$\begingroup$ About the last lines in this remark there is a very nice blog post by Nguyen Tien Zung which summarizes some known results (and showing that in many cases tha type of foliation coming from a $\mathbb R^n$-action is quite limited. zung.zetamu.net/2012/01/rn-actions-on-n-dimensional-manifolds $\endgroup$– Nicola CiccoliCommented Feb 3, 2012 at 17:06
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If i remember correctly the subject was somewhat fashionable in the 60ies and 70ies, when some papers on the subject appeared. I list a couple of references which can give an idea about what was obtained at the time and that can help you in further inquiries:
E. Lima, commuting vector fields on $\mathbb S^3$, Ann. Math. 81, 70-81 1965.
Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46, 467-477 1971.
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$\begingroup$ If i remember correctly the Lima paper contains an answ3r to Smale question:what is the rank of S^3 answrwd by Lima(=1) $\endgroup$ Commented May 7 at 22:04