## On $$\xi$$-conformally flat contact metric manifolds.(English)Zbl 0882.53031

Let $$M$$ be a $$(2n+1)$$-dimensional contact metric manifold with structure $$(\varphi,\xi,\eta,g)$$ and contact form $$\omega$$. $$M$$ is a Sasakian manifold if there exists a normal contact structure on $$M$$. If $$\xi$$ is a Killing vector field, then $$M$$ is called a $$K$$-contact metric manifold. The tangent bundle $$TM$$ can be decomposed as $$TM =\varphi(TM)\oplus{\mathcal L}$$, where $$\mathcal L$$ is the 1-dimensional distribution generated by $$\xi$$. $$M$$ is $$\eta$$-Einstein if there exist functions $$a$$ and $$b$$ such that $$S(X,Y)=ag(X,Y)+b\eta(X)\eta(Y)$$, where $$S$$ is the Ricci tensor field of $$M$$. Let $$C$$ be the Weyl conformal curvature tensor of $$M$$. If the linear operator $$C(X,Y)$$ is an endomorphism of $$\varphi(TM)$$, then $$M$$ is said to be a $$\xi$$-conformally flat contact metric manifold.
In this paper, the authors show that a $$K$$-contact metric manifold $$M$$ is $$\xi$$-conformally flat if and only if it is an $$\eta$$-Einstein Sasakian manifold. The authors give some applications of this theorem.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)