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.


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