On a property of quasi-Kähler manifolds
type: Item
instance of: scholarly article
title: On a property of quasi-Kähler manifolds
author: Galina Anatol'evna Banaru
author: Mihail B. Banaru
published in: Mathematical Notes
publication date: 2024-07-23T00:00:00Z
review text: 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)].
reviewed by: Andreea Olteanu
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 7885067
zbMATH Keywords: almost Hermitian manifolds
zbMATH Keywords: quasi-Kähler manifolds
zbMATH Keywords: almost contact metric manifolds
zbMATH Keywords: quasi-Sasakian manifolds
zbMATH Keywords: \( \eta \)-quasi-umbilical hypersurfaces
MaRDI profile type: Publication
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DOI: 10.1134/S000143462405002X
description: scientific article; zbMATH DE number 7885067