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The structure of a class of $K$-contact manifolds. (English) Zbl 0924.53024

Let $\left(M,g,\phi ,\xi ,\eta \right)$ be a contact metric manifold with a Killing $\xi$ structure vector field (called a $K$-contact manifold) and $C$ its Weyl conformal tensor. Then ${T}_{p}M$, $p\in M$ decomposes into $\phi \left({T}_{p}M\right)\oplus ℒ\left({\xi }_{p}\right)$, where $ℒ\left({\xi }_{p}\right)$ is a 1-dimensional linear subspace of ${T}_{p}M$ generated by ${\xi }_{p}$. It is natural to study the following particular cases:

(i) $C:{T}_{p}M×{T}_{p}M×{T}_{p}M\to ℒ\left({\xi }_{p}\right)$ ,

(ii) $C:{T}_{p}M×{T}_{p}M×{T}_{p}M\to \phi \left({T}_{p}M\right)$,

(iii) $C:\phi \left({T}_{p}M\right)×\phi \left({T}_{p}M\right)×\phi \left({T}_{p}M\right)\to ℒ\left({\xi }_{p}\right)$.

It was shown by the last and first author that in case (i) $M$ is locally isometric to the unit sphere; in case (ii) $M$ is an $\eta$-Einstein Sasakian manifold. This paper shows that in case (iii) if $M$ is compact and ${\phi }^{2}C\left(\phi X,\phi Y\right)\phi Z=0$ (i.e., $M$ is $\phi$-conformally flat), then $M$ is a principal ${S}^{1}$-bundle over an almost Kähler space of constant holomorphic sectional curvature.

##### MSC:
 53C15 Differential geometric structures on manifolds 53C55 Hermitian and Kählerian manifolds (global differential geometry)