On a property of quasi-Kähler manifolds (Q6576753)

The study of structures on hypersurfaces of almost Hermitian manifolds started a long time ago and some intersting results have been established in several papers. In 2020, the authors of the paper under review together with \textit{A. Abu-Saleem} and \textit{L. V. Stepanovo} proved in [Bul. Acad. Științe Repub. Mold. Mat 93(2), 68--75 (2020; Zbl 1478.53040)] proved that if an arbitrary quasi-Kähler manifold satisfies the quasi-Sasakian hypersurfaces axiom, then it is an almost Kähler manifold. This article can be considered as a development of the previous result. \N\NThe authors show in Theorem \(1\) that if a quasi-Kähler manifold satisfies the \(\eta\)-quasi umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kähler manifold. In Theorem \(2\) the authors prove that the quasi-Sasakian structure on an \(\eta\)-quasi-umbilical hypersurface in a quasi-Kähler manifold is either cosymplectic or homothetic to a Sasakian structure. Theorem \(1\) generalizes some classical results of \textit{S. Sasaki} [Tôhoku Math. J. (2) 12, 459--476 (1960; Zbl 0192.27903)] and \textit{H. Yanamoto} [Res. Rep. Nagaoka Tech. College 5, 149--158 (1969; Zbl 0181.50201)], and Theorem \(2\) generalizes findings in [S. Sasaki, loc. cit.; D. E. Blair, J. Differ. Geom. 1, 331--345 (1967; Zbl 0163.43903)].