Generalized symmetric spaces, Yu. P. Solovyov's formula, and the generalized Hermitian geometry
Homogeneous \(\Phi \)-spaces, i.e. homogeneous spaces generated by automorphisms \(\Phi \) of Lie groups, are important objects of differential geometry. They include homogeneous symmetric spaces \((\Phi ^2=\text{Id})\) and homogeneous \(k\)-symmetric spaces \((\Phi ^k= \text{Id})\). There exist homogeneous \(\Phi \)-spaces that are not reductive. Hence, the regular \(\Phi \)-spaces which are contained in the class of reductive spaces play a fundamental role. A distinguishing feature of regular \(\Phi \)-spaces is that each such space has a natural associated object, the commutative algebra \(\mathcal A( \theta )\) of canonical affinor structures. This algebra contains almost complex structures \(J\) \((J^2=-1)\), almost product structures P \((P^2=1)\), \(f\)-structures of K. Yano \((f^3+ f=0)\) and \(h\)-structures \((h^3-h=0)\). In sections 2--4 the author collects basic notions and results of homogeneous regular \(\Phi \)-spaces, specifically, homogeneous \(k\)-symmetric spaces. Yu. P. Solovyov's stimulating influence on this direction during its first stage is illustrated. Moreover, a precise description of all canonical structures of classical type on homogeneous \(k\)-symmetric spaces is given. In sections 5 and 6 the author deals with special canonical \(f\)-structures on regular \(\Phi \)-spaces. Using special canonical \(f\)-structures on homogeneous \(k\)-symmetric spaces with \(k=4n\), \(n \geq 1\), the author presents a new collection of invariant Hermitian \(f\)-structures.
homogeneous \(k\)-symmetric spaces, canonical affinor structures
scientific article; zbMATH DE number 5659093
scientific article