On a property of W4 -manifolds

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.

Keywords

• almost hermitian manifold
• w4 -manifold
• almost contact metric structure
• hypersurface