Homogeneous Hermitian f-manifolds

Abstract

Invariant structures traditionally (and deservedly) play a leading role in the investigation of differential-geometric structures on smooth manifolds. In fact, thanks to a large number of invariant examples many deep results were obtained in the period of stormy development of Hermitian geometry (see [1]–[3], for example), and the flow of research in this area has not slackened even now. At the same time, the study of almost contact structures and their applications, of the f -structures of Yano (f3 + f = 0) and of other generalizations led to the creation in the 1980s in the work of Kirichenko of the elegant concept of generalized Hermitian geometry (see [4], for example). There is a rapidly growing number of papers in various aspects of this new field (see [5], for example), and so the lack of invariant examples is becoming all the more noticeable. There has recently been a qualitative change in the situation, related to the complete solution of the problem of describing canonical structures of classical type on regular Φ-spaces [6]. A rich supply of canonical f -structures has been discovered (including almost complex structures) leading to the determination of large classes of invariant examples in generalized Hermitian geometry (see [7], [8]); this has ensured a continuation of the classical results of Wolf and Gray in Hermitian geometry. The properties of the adjoint Q-algebra [4] form an algebraic basis for distinguishing the most important classes of generalized almost Hermitian structures. In the case of a metric f -structure the composition tensor T was computed explicitly in [4]: