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On the existence of J(4,2)-structures. (English) Zbl 0555.53018

A J(4,2)-structure on a \(C^{\infty}\)-manifold V of dimension \(n=2k\) is a \(C^{\infty}\) (1,1)-tensor field J on V satisfying i) \(J^ 4+J^ 2=0\) and ii) rank J\(=(rank J^ 2+n)/2\). If J is of constant rank V is called J(4,2)-manifold. The operators \(l=-J^ 2\) and \(m=J^ 2+1\) are projectors of the almost product structure on V. Denote by L and M the corresponding distributions. We have \(TV=L+M\). This decomposition is used for proofs of some properties of characteristic classes (Pontjagin classes, Stiefel-Whitney classes and the Euler class in the orientable case) of a J(4,2)-manifold, establishing obstructions to the existence of a J(4,2)-structure on a given manifold.
Reviewer: J.Bureš

MSC:

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