Orbits of Lie Algebra Actions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:53:35Z http://mathoverflow.net/feeds/question/85765 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85765/orbits-of-lie-algebra-actions Orbits of Lie Algebra Actions Zouhair 2012-01-15T20:34:22Z 2012-02-14T09:22:12Z <p>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 $\rho$ ?</p> http://mathoverflow.net/questions/85765/orbits-of-lie-algebra-actions/85892#85892 Answer by Nicola Ciccoli for Orbits of Lie Algebra Actions Nicola Ciccoli 2012-01-17T07:42:53Z 2012-01-17T07:42:53Z <p>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:</p> <p>E. Lima, commuting vector fields on $\mathbb S^3$, Ann. Math. 81, 70-81 1965.</p> <p>Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46, 467-477 1971.</p>