Remarks on the Nijenhuis tensor and almost complex connections.

*(English)*Zbl 0738.53014Given 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.

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.

Reviewer: L.Tamássy (Debrecen)

##### MSC:

53C05 | Connections (general theory) |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |