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.


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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