Welcome to MathOverflow

Question

2

1

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 $\rho$ ?

edited Jan 15 2012 at 22:42

asked Jan 15 2012 at 20:34

Zouhair
21●2

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? – Mariano Suárez-Alvarez Jan 15 2012 at 21:36

(I guess, though, you want the action of $\mathfrak g$ to be free, as in the first sentence?) – Mariano Suárez-Alvarez Jan 15 2012 at 21:51

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. – Zouhair Jan 15 2012 at 22:55

1

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 $Y$. – Robert Bryant Jan 16 2012 at 0:23

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/… – Nicola Ciccoli Feb 3 2012 at 17:06

Nicola Ciccoli · Answer 1 · 2012-01-17 07:42:53Z UTC

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.

Orbits of Lie Algebra Actions

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.