Involutive distribution

Involutive distributions

Given a distribution ${\displaystyle \Delta }$, its sections consist of vector fields on ${\displaystyle M,}$ forming a vector subspace${\displaystyle \Gamma (\Delta )\subseteq \Gamma (TM)={\mathfrak {X}}(M)}$ of the space of all vector fields on ${\displaystyle M}$. (Notation: ${\displaystyle \Gamma (TM)}$ is the space of sections of ${\displaystyle TM.}$) A distribution ${\displaystyle \Delta }$ is called involutive if ${\displaystyle \Gamma (\Delta )\subseteq {\mathfrak {X}}(M)}$ is also a Lie subalgebra: in other words, for any two vector fields ${\displaystyle X,Y\in \Gamma (\Delta )\subseteq {\mathfrak {X}}(M)}$, the Lie bracket${\displaystyle [X,Y]}$ belongs to ${\displaystyle \Gamma (\Delta )\subseteq {\mathfrak {X}}(M)}$.