Differentiable solutions of algebraic equations on manifolds
Samuel I. Goldberg, Nicholas C. Petridis · Kodai Mathematical Journal · 1973
Introduction.Yano [7] introduced the notion of an f-structure, which is a non-null (1,1) tensor field / of constant rank r on a C°° manifold of dimension r+m, satisfying / 3 -f/-0.An almost complex and an almost contact structure are particular cases of an /-structure the existence of an /-structure being equivalent to a reduction of the structural group of the tangent bundle to U(r/2)xO(m).They were studied by various authors ([1], [2], [6], etc.) with particular focus on the case of globally framed structures [2].Extending the concept of an /-structure, Goldberg and Yano [3] introduced the notion of a polynomial structure on a manifold.An /-structure is a particular case of an almost product structure [7], [8].The purpose of this paper is to point out the close relation of the polynomial structures on manifolds and the almost product structures as defined by Walker [8].In § 2 it is shown that any polynomial structure generates an almost product structure.From this follow necessary and sufficient conditions for a distribution to be globally framed and for a manifold to be parallelizable.In §3 reductions of the structural group of the tangent bundle of a polynomial structure are obtained, similar to that for /-structures (see [7]).It is shown that for any polynomial structure with structure polynomial decomposable into distinct irreducible quadratic factors over the reals R that there is an underlying almost complex structure.In §5 an analogue of the normal /-structures [2] is examined which is more general in the sense that the tensor field / is not required to satisfy an algebraic equation. Almost productstructure.Let M be a differentiate manifold.A C°° tensor field / of type (1,1) on M is said to define a polynomial structure if / satisfies the algebraic equation (2.1) P(x)=x m +a m χ n -1 + +a&+a 1 I=Q, where / is the identity mapping and f m ~\P\ f m ~2(p), ~ ,f(P\I are linearly independent for every psM.Clearly, / is non-singular if and only if α^O.The polynomial P(x) is called the structure polynomial.If P(x)=x 2 +I we have an almost complex structure.An almost product structure on a differentiate manifold M is a system of differentiate distributions T lt T 2 , •••, T k such that