Naturally reductive locally 4-symmetric spaces
The class of naturally reductive pseudo-Riemannian 3-symmetric spaces has been determined by \textit{A. Gray} as that of all nearly Kaehler manifolds satisfying an additional (simple) condition for the covariant derivative of the Riemannian curvature [cf. J. Differ. Geom. 7, 343-369 (1972; Zbl 0275.53026)]. As an analogue, the present author characterizes all naturally reductive 4-symmetric spaces as (pseudo) Riemannian manifolds with an f-structure (in the sense of K. Yano) satisfying (i) three identities analogous to the nearly Kaehler property, and (ii) two additional (simple) conditions for the covariant derivative of the curvature. Notice that the number of conditions is larger because, besides the f-structure F satisfying \(F^ 3+F=0\), one has to consider also its associated polynomial structure \(P=F^ 2+I\). Problems of this kind (also for higher orders k) are interesting and far from being obvious.
scientific article; zbMATH DE number 4071844
scientific article