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On some 2-dimensional Hermitian manifolds. (English) Zbl 0603.53033

Let (M,g,J) be a Hermitian manifold of complex dimension n and \(\Omega (X,Y)=g(X,JY)\) the Kähler form. Recall that g is called locally conformal Kähler (l.c.K.) if one can find an open covering \(M=\cup_{\alpha}U_{\alpha}\) such that \(g| U_{\alpha}=e^{\sigma_{\alpha}}g_{\alpha}\), where \(\sigma_{\alpha}\) is a differentiable function and \(g_{\alpha}\) is a Kähler metric on \(U_{\alpha}\). Let \(\Omega_{\alpha}\) be the Kähler form of \(g_{\alpha}\), \(\Omega_{\alpha}=e^{- \sigma_{\alpha}}\Omega\). The condition \(d\Omega_{\alpha}=0\) is equivalent to \(d\Omega =\omega \wedge \Omega\) with \(\omega | U_{\alpha}=d\sigma_{\alpha}\). From this identity one obtains \(\omega =(1/(n-1))i(\Omega)d\Omega\). Since \(\omega =d\sigma_{\alpha}\), one has \(d\omega =0.\)
Suppose now that g is not necessarily l.c.K. and consider the 1-form \(\omega\) obtained from \(\Omega\) by the above expression. If \(\dim_ C M=2\) then one obtains easily that \(d\Omega =\omega \wedge \Omega\). \(\omega\) is called the Lee form of the Hermitian 2-manifold. \(d\omega =0\) is equivalent to say that M is l.c.K. The aim of the paper is to discuss 2-manifolds defined by other interesting properties of the Lee form \(\omega\). For this purpose the authors use a classification method due to A. Gray and L. M. Hervella [Ann. Mat. Pura Appl., IV. Ser. 123, 35-58 (1980; Zbl 0444.53032)]. For a unitary vector space (V,g,J) of real dimension 4 they consider the vector space \(\tau\) of the 2-covariant tensors having the same ”symmetries” as \(\nabla \omega\). They write a (non-unique) decomposition of this space as sum \(\tau =\tau_ 1\oplus \tau_ 2\oplus \tau_ 3\oplus \tau_ 4\oplus \tau_ 5\) of 5 irreducible components of the natural action of U(2). In this way they obtain the following 5 classes of Hermitian manifolds:
1) \((\nabla_ x\omega)(Y)=\lambda g(X,Y)\). 2) \(d\omega =0\), \(\delta \omega =0\), \(L_ Ag=0\) (where \(A=-JB\) and B is such that \(g(B,X)=\omega (X)\), \(\forall X)\). 3) \(d\omega =0\), \(L_ B\Omega =0\). 4) \(L_ Bg=0\), \(L_ B\Omega =0\). 5) \(L_ Bg=0\), \(L_ Ag=0.\)
The l.c.K. case is characterized by \(\nabla \omega \in \tau_ 1\oplus \tau_ 2\oplus \tau_ 3\). They prove, for example, that when M is compact and belongs to the classes (1), (2), (5) then one has necessarily \(\nabla \omega =0\) (generalized Hopf manifolds) and find examples, when M is not compact, of non-generalized Hopf manifolds belonging to classes (3) and (4).
Reviewer: J.Girbau

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B35 Local differential geometry of Hermitian and Kählerian structures

Citations:

Zbl 0444.53032
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References:

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