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Remarks on the Nijenhuis tensor and almost complex connections. (English) Zbl 0738.53014
Given a (1,1)-tensor field $$S$$ the author determines all natural (1,2)- tensor fields of the same type as the Nijenhuis tensor $$N_ S$$. He shows the nonexistence of affine connections polynomially naturally induced from $$S$$. Also all connections $$\tilde\nabla$$ naturally induced from a given symmetric affine connection and from $$S$$ such that $$\hbox{Tor }\tilde\nabla=\lambda N_ S$$ ($$\lambda\in R$$) are found; and conditions under which these $$\tilde\nabla$$ are almost complex connections are deduced.
The paper is related to and has been motivated by a result of S. Kobayashi and K. Nomizu [Foundations of differential geometry. Vol. II. (Moskva: “Nauka” 1981; Zbl 0526.53001)] giving for every almost complex manifold with an almost complex structure $$J$$ an almost complex affine connection $$\tilde\nabla$$ such that $$\hbox{Tor }\tilde\nabla={1\over 8}N_ J$$. The paper under review shows that besides Kobayashi and Nomizu’s example there are still many naturally induced solutions.

##### MSC:
 53C05 Connections (general theory) 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Zbl 0526.53001
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