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Summary:
The author considers the Nijenhuis map assigning to two type (1,1) tensor fields $\alpha$, $\beta$ a mapping $$\langle \alpha, \beta\rangle : (\xi, \zeta) \mapsto [\alpha(\xi), \beta(\zeta)] + \alpha \circ \beta ([\xi, \zeta]) - \alpha([\xi, \beta(\zeta)]) - \beta([\alpha(\xi), \zeta)]),$$ where $\xi$, $\zeta$ are vector fields. Then $\langle \alpha, \beta\rangle$ is a type (2,1) tensor field (Nijenhuis tensor) if and only if $[\alpha, \beta] = 0$. Considering a smooth manifold $X$ with a smooth action of a Lie group, a secondary invariant may be defined as a mapping whose area of invariance is restricted to the inverse image of an invariant subset of $X$ under another invariant mapping. The author recognizes a secondary invariant related to the above Nijenhuis tensor and gives a complete list of all secondary invariants of similar type. In this way he proves that all bilinear natural operators transforming commuting pairs of type (1,1) tensor fields to type (2,1)!
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