I was reading through "Lectures on Symplectic Geometry" by Ana Cannas da Silva.
In Chapter 10 where the book introduces contact structures, it says $ker \ (d\alpha_p)$ is one dimensional where $\alpha$ is the contact form. I do not see why this is true. Thanks for any help.
In that context, we postulate that $d\alpha_p$ is non-degenerate on a hyperplane $H$ (which implies that $H$ is even dimensional). This implies that considered as a form on ambient tangent space $d\alpha_p$ can not have kernel of dimension higher than 1 -- otherwise that kernel will intersect the hyperplane non-trivially, leading to a contradiction. On the other hand, being an anti-symmetric form on an odd dimensional space $d\alpha_p$ must have a non-trivial kernel. So the kernel is one dimensional.