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Zbl 0924.53024
Cabrerizo, J.L.; Fernández, L.M.; Fernández, M.; Guo Zhen
The structure of a class of $K$-contact manifolds. (English)
[J] Acta Math. Hung. 82, No.4, 331-340 (1999). ISSN 0236-5294; ISSN 1588-2632/e

Let $(M,g,\phi,\xi,\eta)$ 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 $\phi(T_p M)\oplus \Cal L(\xi_p)$, where $\Cal L(\xi_p)$ is a 1-dimensional linear subspace of $T_pM$ generated by $\xi_p$. It is natural to study the following particular cases: \par (i) $C:T_p M\times T_p M\times T_p M\to \Cal L(\xi_p)$ ,\par (ii) $C:T_pM\times T_p M \times T_p M\to \phi (T_pM)$, \par (iii) $C:\phi(T_pM)\times \phi (T_pM)\times \phi (T_pM)\to \Cal L(\xi_p)$. \par 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 $\phi^2C(\phi X,\phi Y)\phi Z=0$ (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.
[Lajos Tamássy (Debrecen)]

MSC 2000:: *53C15 Geometric structures on manifolds
53C55 Complex differential geometry (global)

Keywords: $K$-contact structures; conformally flat; almost Kähler

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