Abstract
The properties of almost Hermitian manifolds belonging to the Gray — Hervella class W4 are considered. The almost Hermitian manifolds of this class were studied by such outstanding geometers like Alfred Gray, Izu Vaisman, and Vadim Feodorovich Kirichenko. Using the Cartan structural equations of an almost contact metric structure induced on an arbitrary oriented hypersurface of a W4-manifold, some results on totally umbilical and totally geodesic hypersurfaces of W4-manifolds are presented. It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manifold is either homothetic to a Sasakian structure or cosymplectic. Moreover, the quasi-Sasakian structure is cosymplectic if and only if the hypersurface is a totally geodesic submanifold of the considered W4-manifold. From the present result it immediately follows that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a locally conformal Kählerian (LCK-) manifold also is either homothetic to a Sasakian structure or cosymplectic.
Highlights
Данная статья продолжает работу автора, связанную с изучением почти контактных метрических структур на гиперповерхностях W4 -многообразий [2,3,4,5]
It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manifold is either homothetic to a Sasakian structure or cosymplectic
From the present result it immediately follows that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a locally conformal Kählerian (LCK-) manifold is either homothetic to a Sasakian structure or cosymplectic
Summary
Данная статья продолжает работу автора, связанную с изучением почти контактных метрических структур на гиперповерхностях W4 -многообразий [2,3,4,5]. 1. Значение класса W4 почти эрмитовых многообразий определяется прежде всего тем, что этот класс содержит все локально конформные келеровы (locally conformal Kählerian, LCK-) многообразия. Банару состоящую из почти комплексной структуры J и римановой метрики g , , причем J и g , должны удовлетворять условию JX , JY X , Y , X ,Y (M 2n ), где (M 2n ) — модуль гладких векторных полей на многообразии M 2n . Почти эрмитова структура на многообразии M 2n принадлежит классу W4 , если
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