Let $(M,\xi)$ be a coorientable contact $(2n+1)$-manifold with $\alpha$ and $R$ as contact form and Reeb vector field, respectively. A contact metric structure on $(M,\xi)$ is a pair $(J,g)$ where $J$ is a (1,1)-tensor satisfying $$J^2=-Id+\alpha\otimes R,$$ and $g$ is a metric satisfying $d\alpha(-,-)=g(J-,-)$ and $g(-,-)=g(J-,J-)+\alpha^2(-,-)$.
I have a couple of probably simple questions about this definition:
- What is the area form of $g$? is it true that it is simply $\alpha\wedge(d\alpha)^n$ and that $\star\alpha=(d\alpha)^n$ for the hodge star operator?
- Can we extend $J$ to forms? In turn, what is its relation with $\star$, i.e. $[J,\star]$=?
If it makes it easier, I'm mostly interested in the 3-dimensional case, where $J$ on $\xi$ is the rotation by $\pi/2$ in the positive direction.