An interesting class of contact manifolds is the class of $K$-contact manifolds ($\mathcal{L}\_\xi g=0$) which have been studied by many authors. It is natural to study conformal Killing-contact manifolds. This means that the characteristic vector field satisfies the conformal killing equation; i. e. $\mathcal{L}\_\xi g=\sigma g$ where $\sigma$ is a smooth function on $M$. Why is this structure not studied? 



Conformal Killing vector field on contact manifolds
The reason is that the g-trace of the conformal Killing equation gives 2(divergence of characteristic vector field)=n(sigma). But divergence of characteristic vector field on a contact metric manifold is zero. Hence, sigma = 0. Thus it reduces to the K-contact manifold. 


