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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.

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