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Differential Geometry of Manifolds

2025 №56

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Abu-Saleem A. Banaru M.B. Banaru G.A.

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

1. Abass, M. Y.: On generalized Kenmotsu manifolds as hypersurfaces of Vaisman — Gray manifolds. Vladikavkaz. Math. J., 26:1, 5—12 (2024).

2. Kirichenko, V. F., Stepanova, L. V.: The geometry of hypersurfaces of quasi-Kählerian manifolds. Russian Math. Surveys, 50:2, 440—441 (1995).

3. Stepanova, L. V.: Contact geometry of hypersurfaces of quasi-Käh­le­rian manifolds. PhD thesis. Moscow State Pedagogical University V. I. Le­nin (1995).

4. Banaru, M. B.: Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra. J. Harbin Inst. Technol. (N. S.). 8:1, 38—40 (2001).

5. Banaru, M. B.: On Sasakian hypersurfaces in 6-dimensional Her­mi­tian submanifolds of the Cayley algebra. Sb. Math., 194:8, 1125—1136 (2003).

6. Banaru, M. B.: The type number of the cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of the Cayley algebra. Siberian Math. J., 44:5, 765—773 (2003).

7. Banaru, M. B.: On skew-symplectic hypersurfaces of six-dimen­sio­nal Kählerian submanifolds of the Cayley algebra. Russian Math. (Iz­vestia Vuzov), 47:7, 60—63 (2003).

8. Abu-Saleem, A., Banaru, M. B.: On almost contact metric hypersur­fa­ces of nearly Kählerian 6-sphere. Malaysian J. of Math. Sci., 8:1, 35—46 (2014).

9. Banaru, M. B.: On minimality of a Sasakian hypersurface in a -ma­nifold. Saitama Math. J. 20, 1—7 (2002).

10.  Abu-Saleem, A., Banaru, M. B.: Two theorems on Kenmotsu hy­per­surfaces in a -manifold. Studia Univ. Babeş — Bolyai. Math. 51:3, 3—11 (2005).

11.  Banaru, M. B., Kirichenko, V. F.: Almost contact metric structures on the hypersurface of almost Hermitian manifolds. J. Math. Sci., New York. 207:4, 513—537 (2015).

12.  Stepanova, L. V., Banaru, G. A., Banaru, M. B.: On quasi-Sasakian hypersurfaces of Kählerian manifolds. Russian Math. (Izvestia Vuzov), 80:1, 73—75 (2016).

13.  Banaru, M. B., Banaru, G. A.: On hypersurfaces with Kirichenko — Uskorev structure in Kählerian manifolds. Sib. Elektron. Math. Izv., 17, 1715—1721 (2020).

14.  Banaru, G. A.: On nonexistence of Kenmotsu structure on acm-hy­persurfaces of cosymplectic type of a Kählerian manifold. DGMF, 50, 23—28 (2019).