Suppose $M$ is a manifold of dimension $2m+1$. Let $\xi$ be a sub bundle of the tangent bundle of rank $2m$. We also call this sub bundle a distribution. I want to show that locally there exists a integrable submanifold of dimension $m$. The reason I am trying to prove this is because I have heard contact structures are as far as they can be from being integrable and yet they have this property. So I am wondering if this minimum is always achieved.
My try:
First since the question is local we can just think of the trivial bundle case. Then if we can find a $m-$dimensional subbundle of the of distribution which is involutive we would be done using Frobenius theorem. Now this is where I am stuck that is I am unable to find such a sub bundle. Any hints are appreciated. Thank you.
I got to know a solution from my professor.
We are only looking locally so we consider a section of normal bundle of the distribution say $\eta$. Now we define a form on the distribution given by $[X,Y]=\alpha(X,Y)\eta$. Now clearly this is alternating. Now if this is degenerate let $\kappa$ be the Kernel of dimension $2k$. Then consider $\xi/\kappa$, then $\alpha$ is symplectic on this. So now we know there is a symplectic basis $e_1,e_2,...,e_{2m-2k}$ such that $\alpha$ is zero on $e_1,..,e_{m-k}$. This precisely means the lie bracket lies in the distribution spanned by these vectors. Now adding $\kappa$ we get that there is a involutive sub distribution of rank $m+k$. The worst case happens when $k=0$ in which case the sub distribution will have rank only $m$.