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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)! (English) |
