Coefficients of the Lie brackets of a set of vector fields

(see also about solvable algebras and solvable structures)

Let $M$ be a manifold and $X_1,\dots ,X_r$ vector fields on $M$. The condition

where $f_{ij}^k$ are functions, implies that the distribution $\mathcal{S}(\{X_1,\dots ,X_r\})$ generated by them is involutive. Frobenius theorem says that there exist integral manifolds.

But there is a more restricted condition, constant coefficients:

where $c_{ij}^k\in \mathbb{R}$. This means that they constitute a finite dimensional Lie subalgebra of $\mathfrak{X}(M)$ (with (1) they constitute a possibly infinite dimensional one). In this case there exists a finite dimensional Lie group $G$ acting on $M$ such that the integral manifolds of the distribution $\mathcal{S}(\{X_1,\dots ,X_r\})$ are the orbits of $G$!!!

And even more, if every

then the group is isomorphic to the translations ${\mathbb{R}}^r$.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es