I would greatly appreciate it if you could help me with the following:
I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as the zero set of:
$dw(R_w ,. )=0$
(where $R_w$ is known to exist by Linear Algebra, since manifold is odd-dimensional )
Or does one some how use the fact that the Lie derivative of$ w$ with respect to $R_w$ is zero (or something else)?
I know that a Reeb field is "stronger"than a plain contact field, in that a contact field preserves just the contact structure, while a Reeb field preserves the form itself, i.e., if $\phi$ is the flow of $R_w$ , then $\phi^*w$=$gw$ for contact fields (i.e., the kernel of the form gw is the same as kerw), but $\phi^*w=w$ for Reeb fields.
Hopefully, too, someone could help me figure out how to show that a contact vector field ( one whose flow preserves the contact structure) , that is transverse to the contact pages is a Reeb field. I'm kind of clueless here.
Thanks for any help, suggestions,