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?
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1 Answer
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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.
