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Zouhair
Zouhair
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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 $\rho$ ?

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. 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 Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$, i.e., the leaves of $\mathcal{F}$ are the $\mathfrak{g}$-orbits of $\rho$ ?

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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Zouhair
Zouhair
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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 distrubitiondistribution of constant rank. 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 Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$, i.e., the leaves of $\mathcal{F}$ are the $\mathfrak{g}$-orbits of $\rho$ ?

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 distrubition of constant rank. 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 Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$, i.e., the leaves of $\mathcal{F}$ are the $\mathfrak{g}$-orbits of $\rho$ ?

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. 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 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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Zouhair
Zouhair
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Orbits of Lie Algebra Actions

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 distrubition of constant rank. 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 Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$, i.e., the leaves of $\mathcal{F}$ are the $\mathfrak{g}$-orbits of $\rho$ ?