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Canonical affinor structures of classical type on regular $$\Phi$$-spaces. (English. Russian original) Zbl 0872.53025
Sb. Math. 186, No. 11, 1551-1580 (1995); translation from Mat. Sb. 186, No. 11, 3-34 (1995).
Let $$G$$ be a connected Lie group, $$H$$ its Lie subgroup. The homogeneous space $$G/H$$ is called a $$\Phi$$-space if $$(G^{\Phi})^{\circ}{\i}H{\i}G^{\Phi}$$, where $$\Phi$$ is an automorphism of $$G$$. The $$\Phi$$-space is regular if the Lie algebras $$g, h$$ of $$G, H$$ satisfy $$g = h \oplus\operatorname{Ker}(\varphi - \operatorname{id})$$, where $$\varphi = d\Phi_e$$. An invariant affinor structure on $$G/H$$ is determined by its value at the point $$eH$$, which is an arbitrary linear operator $$\xi$$ on $$m = \operatorname{Ker}(\varphi - \operatorname{id})$$ commuting with $$\operatorname{Ad}H$$. Such a structure is called canonical if $$\xi$$ is a polynomial in $$\theta = \varphi|m$$. The goal of the paper is to describe canonical structures on an arbitrary $$\Phi$$-space that are almost complex, almost product or $$f$$-structures, i.e., are determined by an operator $$\xi$$ satisfying $$\xi^2 = \pm 1$$ or $$\xi^3 +\xi = 0$$. This reduces to solving the equations $$x^2 = \pm 1, x^3 +x = 0$$ in a quotient of the ring $$\mathbf R[\lambda]$$ modulo a non-zero ideal. A procedure for solving these equations is proposed. It is proved, in particular, that on a regular $$\Phi$$-space $$G/H$$ there exist precisely $$2^s$$ canonical almost complex structures and $$3^s-1$$ canonical $$f$$-structures, where $$s$$ is the number of distinct pairs of complex roots of the minimal polynomial of $$\theta$$. The case when $$\Phi$$ is periodical is studied in more detail.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces
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