2025 №56

On minimality of almost contact metric hypersurfaces in almost Hermitian manifolds

Abstract

We consider obtained in 2024 by M. Y. Abass a minimality criterion for hypersurfaces with classical generalized Kenmotsu structure in a Vaisman — Gray manifold. This condition is known to have arisen in the study of hypersurfaces equipped with certain kinds of almost contact metric structures in almost Hermitian manifolds belonging to various Gray — Hervella classes. The con­di­tion was sometimes a minimality criterion for an almost con­tact met­ric hypersurface of an almost Hermitian manifold; in other ca­ses, it turned out to be only necessary or only sufficient.

In the present note, we formulate two problems:

1) to analyze in detail the above-mentioned minimality con­di­tion for an almost contact metric hypersurface of an almost Her­mi­tian manifold;

2) to find out how the Abass minimality condition for a hy­per­sur­face with the classical generalized Kenmotsu structure in a Vais­man — Gray manifold is related to the minimality of a hyper­sur­face with the Kirichenko — Uskorev structure, which is also a ge­ne­ralization of the Kenmotsu structure.

Reference

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