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Some remarks on polynomial structures. (English) Zbl 0974.53021

Author’s abstract: A polynomial structure \((M,f)\) of degree \(n\) on a connected \(C^\infty\)-manifold \(M\) is a (1,1)-tensor field \(F\) satisfying on \(M\) an equation \[ p(F)=F^n+ a_1F^{n-1}+ \cdots+a_{n-1} F+a_nI=0 \] where the coefficients of the structural polynomial \(p\) are either constants or functions. The integrability of polynomial structures (with constant coefficients) the structure polynomial of which has only simple roots were investigated in [J. Vanzura, Kodai Math. Sem. Rep. 27, 42-50 (1976; Zbl 0326.53050)]. We will analyze here integrability conditions found there in more details, using a complexification of the tangent bundle of the base manifold. We will associate with \((M,F)\) a complex almost product structure on the complexification of the tangent bundle, and will prove that \((M,F)\) is integrable if and only if its associated complex almost product structure is integrable.

MSC:

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

Citations:

Zbl 0326.53050
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References:

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