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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.)
Zbl 0326.53050
Full Text:
##### References:
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