A couple of basic questions on contact structures

Question

Consider the following definition of contact structures:

A contact structure on an n dimensional smooth manifold is a codimension 1 tangent distribution $\xi$ whose first curvature $\beta:\xi\times \xi\to TM/\xi$, defined by $\beta_p:=[-,-]_p\mod \xi_p$, is a non singular at every point.

1. Can we automatically infer that the manifold is odd-dimensional? What is the proof of that?

2. Can we define contact structures simply as bracket-generating codimension 1 distributions? If the distribution is weakly regular, meaning that the Lie flag consists of subbundles, it seems an equivalent definition. Am I wrong?

Answer 1

Question 1 is essentially answered in the comment by @Didier : It is a fact of linear algebra that non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional spaces. (Basically, it is clear that there is no non-zero skew symmetric bilinear form on $\mathbb R$.) Starting from a non-degenerate, skew symmetric bilinear form $b$ on a vector space $V$ is is easy to see that one can write $V=V_1\oplus V_2$, where $\dim(V_1)=2$ and the restriction of $b$ to both subspaces is non-degenerate and this implies the claim.

For question 2, the answer is yes in dimension 3 (if you include weak regularity) and no in higher dimensions. The fact that $\xi$ is bracket generating and weakly regular just means that $\beta_p\neq 0$ for any $p\in M$. In dimension 3, this implies non-degeneracy (because skew symmetric bilinear forms have even rank), in higher dimensions it doesn't.