and $-i$, and the corresponding eigenspaces will be denoted by $T^{1,0}$ and $T^{0,1}$, respectively. The integrability condition can then be reformulated as $$X,Y\in T^{1,0} \text{ implies } [X,Y]\in T^{1,0}.$$

Now consider a closed $2n+1$-manifold $M$ with a cooriented contact structure $\xi=\ker \theta$. This means that $\theta \wedge d\theta^{n} \neq0 $. The **Reeb vector field** of $\theta$ is the vector field $T$ uniquely determined by the equations $${\theta}(T)=1,\quad \text{and}\quad d{\theta}(T,{\cdot})=0.\label{eq:defining_Reeb}$$

A **pseudohermitian manifold** is a triple $(M^{2n+1},\theta,J),$ where $\theta$ is a contact form on $M$ and $J$ is a CR structure on $\ker \theta$. The **Levi form** $\left \langle \ ,\  \right \rangle $ is the Hermitian form on $T^{1,0}$ defined by $$H(Z,W)=\left \langle Z,W\right \rangle =-i\left \langle d\theta,Z\wedge
\overline{W}\right \rangle .$$

We can extend this Hermitian form $\left \langle \ ,\  \right \rangle $ to $T^{0,1}$ by defining $\left \langle \overline{Z},\overline{W}\right \rangle
=\overline{\left \langle Z,W\right \rangle }$ for all $Z,W\in T^{1,0}$. Furthermore, the Levi form naturally induces a Hermitian form on the dual bundle of $T^{1,0}$, and hence on all induced tensor bundles.

We now restrict ourselves to **strictly pseudoconvex** CR manifolds, or in other words compatible complex structures $J$. This means that the Levi form induces a Hermitian metric 