Assume that $M$ is an arbitrary manifold.

Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?

If not, what is a counter example?

In particular what is the answer to this question for $M=\mathbb{R}^{2}$ or $M=S^{2}$?

The question is a particular case of the following general question:

Question: For a $n$ dimensional manifold $M$, is it true to say that the minimal codimension of Lie sub algebras of $\chi^{\infty}(M)$ is equal to $n$?

The above question serachs for obstruction for finite codimensionality of Lie subalgebras (codimension equal to one) of $\chi^{\infty}(M)$ while the following post concerns diversity and variation of finite dimensional Lie subalgebras.

A Manifold for which $\chi^{\infty}(M)$ is rich

Assume that $M$ is an arbitrary manifold.

Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?

If not, what is a counter example?

In particular what is the answer to this question for $M=\mathbb{R}^{2}$ or $M=S^{2}$?

The question is a particular case of the following general question:

Question: For a $n$ dimensional manifold $M$, is it true to say that the minimal codimension of Lie sub algebras of $\chi^{\infty}(M)$ is equal to $n$?

The above question serachs for obstruction for finite codimensionality of Lie subalgebras (codimension equal to one) of $\chi^{\infty}(M)$ while the following post concerns diversity and variation of finite dimensional Lie subalgebras.

A Manifold for which $\chi^{\infty}(M)$ is rich

Assume that $M$ is an arbitrary manifold.

Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?

If not, what is a counter example?

In particular what is the answer to this question for $M=\mathbb{R}^{2}$ or $M=S^{2}$?

The above question serachs for obstruction for finite codimensionality of Lie subalgebras (codimension equal to one) of $\chi^{\infty}(M)$ while the following post concerns diversity and variation of finite dimensional Lie subalgebras.

A Manifold for which $\chi^{\infty}(M)$ is rich

Assume that $M$ is an arbitrary manifold.

Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?

If not, what is a counter example?

In particular what is the answer to this question for $M=\mathbb{R}^{2}$ or $M=S^{2}$?

The question is a particular case of the following general question:

Question: For a $n$ dimensional manifold $M$, is it true to say that the minimal codimension of Lie sub algebras of $\chi^{\infty}(M)$ is equal to $n$?

The above question serachs for obstruction for finite codimensionality of Lie subalgebras (codimension equal to one) of $\chi^{\infty}(M)$ while the following post concerns diversity and variation of finite dimensional Lie subalgebras.

A Manifold for which $\chi^{\infty}(M)$ is rich