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Pseudoconnections on an almost complex manifold. (English) Zbl 0806.53031
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 447-456 (1992).
Let \((M,J)\) be an almost complex manifold and \((\nabla,A)\) a linear pseudoconnection on \(M\), where \(A\) is a tensor field of type (1,1). The existence of a torsion-free almost complex pseudoconnection on \(M\) has been characterized by other authors. In this paper, the author gives an outline of an alternative proof of the same result. He constructs a tensor field \(L^ J_ A\) depending on \(A\) and \(J\) such that \(L^ J_ A = 0\) is a necessary and sufficient condition for the existence of a torsion-free almost complex pseudoconnection \((\nabla, A)\). He gives also some special examples. Furthermore, he considers a hermitian metric on \((M,J)\) and the Levi-Civita pseudoconnection \((\nabla,A)\). He proves that if \(L^ J_ A = 0\) holds, then \((\nabla,A)\) is almost complex if and only if \(d_ A \Phi = 0\), where \(\Phi\) is the fundamental 2-form on \((M,J,g)\). Finally, the author defines a twisted Kähler manifold as a quadruple \((M,J,g,A)\) with \(L^ J_ A = 0\) and \(d_ A \Phi = 0\), and he gives examples of such manifolds.
For the entire collection see [Zbl 0764.00002].
Reviewer: A.M.Pastore (Bari)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C05 Connections (general theory)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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