Canonical affinor structures of classical type on regular \(\Phi\)-spaces. (English. Russian original) Zbl 0872.53025

Balashchenko, V. V.; Stepanov, N. A.

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.

Reviewer: A.L.Onishchik (Yaroslavl)

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

Keywords:

regular \(\Phi\)-space; canonical affinor structure; almost complex structure; almost product structure; \(f\)-structure