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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)
##### MSC:
 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