Search
Submit an article
Differential Geometry of Manifolds

2023 №54(1)

Back to the list Download the article
Banaru M.B.

A note on ߟ-quasi-umbilical hypersurfaces in almost Hermitian manifolds

DOI
10.5922/0321-4796-2023-54-1-3
Pages
23-28
almost Hermitian manifold almost contact metric struc­ture -quasi-umbilical hypersurface minimal hypersurface

Abstract

In the present note, we consider the introduced by Lidia Vasil’evna Stepanova notion of an -quasi-umbilical hypersurface in an almost Her­mitian manifold. We show that the notion of an -quasi-umbilical hyper­surface in an almost Hermitian manifold is connected with the notion of a minimal hypersurface in this manifold.

Using the classical theory of minimal hypersurfaces in Riemannian manifolds and Kirichenko — Stepanova general theory of almost contact metric hypersurfaces in almost Hermitian manifolds, we establish that an -quasi-umbilical hypersurface of a nearly Kählerian manifold is minimal if and only if this hypersurface is totally umbilical.

Taking into account the connection between the notions of a minimal hypersurface and of an -quasi-umbilical hypersurface in an almost Her­mitian manifold, we conclude that some well-known results in the theory of almost contact metric hypersurfaces in almost Hermitian manifolds can be reformulated.

The problem of the existence of a non-umbilical minimal -quasi-umbilical hypersurface of a quasi-Kählerian manifold is posed.