ON HYPERSURFACES OF MANIFOLDS EQUIPPED WITH A HYPERCOSYMPLECTIC 3-STRUCTURE

Abstract

Hypersurfaces of a Riemannian manifold equipped with a hypercosymplectic 3-structure are studied. Integrability condi- tions for certain distributions on the hypersurface are investigated. Geometry of leaves of certain distribution are also studied. The quaternionic analog of almost complex structures is the almost quaternion (hypercomplex) which is defined by three local (global) al- most complex structures which satisfy the quaternionic relations as the imaginary quaternions satisfy ((3)). Quaternion Kahler manifolds and hyper-Kahler manifolds are special and interesting cases of Riemannian manifolds with almost quaternion and almost hypercomplex structure, respectively. Quaternion Kahler manifolds are Einstein, while hyper- Kahler manifolds are Ricci flat. An almost contact 3-structure was defined by Kuo ((5)) and it is closely related to both almost quaternion and almost hypercomplex structures. Hypersurfaces of manifolds with almost hypercomplex struc- ture inherit naturally three almost contact structures which constitute an almost contact 3-structure. An almost contact metric 3-structure manifold is always (4m + 3)-dimensional. The structural group of the tangent bundle of a (4m + 3)-dimensional manifold equipped with an almost contact 3-structure is reducible to Sp(m) £ I3. In particular, if each almost contact metric structure of an almost contact metric 3-structure is Sasakian, then this structure is called a Sasakian 3-structure. Riemannian manifolds with Sasakian 3-structure