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

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 \(\phi(T_p M)\oplus \mathcal L(\xi_p)\), where \(\mathcal L(\xi_p)\) is a 1-dimensional linear subspace of \(T_pM\) generated by \(\xi_p\). It is natural to study the following particular cases:
(i) \(C:T_p M\times T_p M\times T_p M\to \mathcal L(\xi_p)\) ,
(ii) \(C:T_pM\times T_p M \times T_p M\to \phi (T_pM)\),
(iii) \(C:\phi(T_pM)\times \phi (T_pM)\times \phi (T_pM)\to \mathcal L(\xi_p)\).
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.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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