On nonexistence of Kenmotsu structure on аст-hypersurfaces of cosymplectic type of a Kählerian manifold

Abstract

Almost contact metric (аст-)structures induced on oriented hypersur­fa­ces of a Kählerian manifold are considered in the case when these аст-struc­tures are of cosymplectic type, i. e. the contact form of these struc­tu­res is closed. As it is known, the Kenmotsu structure is the most important non-trivial example of an almost contact metric structure of cosymplectic type. The Cartan structural equations of the almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold are obtai­ned. It is proved that an almost contact metric structure of cosymplectic ty­pe on a hypersurface of a Kählerian manifold of dimension at least six can­not be a Kenmotsu structure. Moreover, it follows that oriented hyper­sur­faces of a Kählerian manifold of dimension at least six do not admit non-trivial almost contact metric structures of cosymplectic type that be­long to any well studied class of аст-structures. The present results gene­ralize some results on almost contact metric structures on hypersurfaces of an almost Hermitian manifold obtained earlier by V. F. Kirichenko, L. V. Stepanova, A. Abu-Saleem, M. B. Banaru and others.

Keywords

• kählerian manifold
• almost contact metric structure
• struc­ture of cosymplectic type
• kenmotsu structure