Mixed 3-Sasakian structures and curvature

Abstract

Abstract. We deal with two classes of mixed metric 3-structures, namely the mixed 3-Sasakian structures and the mixed metric 3-contact structures. First, we study some properties of the curvature of mixed 3-Sasakian structures. Then we prove the identity between the class of mixed 3-Sasakian structures and the class of mixed metric 3-contact structures.

Semi-Riemannian hypersurfaces in manifolds with metric mixed 3-structures

“…It is called mixed 3-structure, which appears in a natural way on lightlike hypersurfaces in paraquaternionic manifolds: such hypersurfaces inherit naturally two almost paracontact structures and an almost contact structure, satisfying analogous conditions to those satisfied by almost contact 3-structures [11]. This concept has been recently refined in [6], where the authors have introduced positive and negative mixed metric 3-structures. A compatible metric with a mixed 3-structure is necessarily semi-Riemann and mixed 3-Sasakian manifolds are Einstein [6,9], hence the possible importance of these structures in theoretical physics (see also [8]).…”

“…This concept has been recently refined in [6], where the authors have introduced positive and negative mixed metric 3-structures. A compatible metric with a mixed 3-structure is necessarily semi-Riemann and mixed 3-Sasakian manifolds are Einstein [6,9], hence the possible importance of these structures in theoretical physics (see also [8]). In this article, combining the techniques of semi-Riemannian geometry [13] and the techniques of foliations theory [3] with the techniques of hypersurface theories in manifolds equipped with almost contact metric 3-structures [2,16], we start the study of semi-Riemannian hypersurfaces of co-index both 0 and 1 in a manifold endowed with a mixed 3-structure and a compatible metric.…”

“…We also note that the main property of a manifold endowed with a mixed 3-Sasakian structure is given by the following theorem (see [6,9]): Theorem 2.4. Any manifold endowed with a mixed 3-Sasakian structure is an Einstein space.…”

“…We also note that the negative metric mixed 3-structures on manifolds have been recently investigated in [8,9]. Definition 2.3 [6]. Let M , (φ α , ξ α , η α ) α=1,3 , g be a manifold with a metric mixed 3-structure.…”

“…Definition 2.2 [6]. A mixed 3-structure on a smooth manifold M is a triple of structures (φ α , ξ α , η α ), α ∈ {1, 2, 3}, which are almost paracontact structures for α = 1, 2 and almost contact structure for α = 3, satisfying the following conditions:…”