zbMATH — the first resource for mathematics

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.

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