In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds find necessary sufficient conditions under which these are homothetic to Sasakian manifolds. First, four results in paper deal with finding on a manifold be manifold, the fifth result deals condition manifold. Finally, simply Einstein

It is proved that a Riemannian manifold M isometrically immersed in a Sasakian space form˜M(c) of constant ϕ-sectional curvature c < 1, with the structure vector field ξ tangent to M, satisfies Chen's basic equality if and only if it is a 3-dimensional minimal invariant submanifold. 1. Introduction. Let˜M be an m-dimensional almost contact manifold endowed with an almost contact structure (ϕ,ξ,...

We prove the following results: (i) A Sasakian metric as a nontrivial Ricci soliton is null η-Einstein, and expanding. Such a characterization permits to identify the Sasakian metric on the Heisenberg group H as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an ηEinstein contact metric manifold M has a vector field V leaving the structure tensor and the scalar curvatur...

We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.

It is well-known that 7-dimensional 3-Sasakian manifolds carry a oneparametric family of compatible G2-structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G2-structure in this family whose characteristic connection ∇ c along with its parallel spinor field Ψ0 can be used for a thorough investigation of t...

It is still an open problem whether Riemannian manifolds all of whose local geodesic symmetries are volume–preserving (i.e., D’Atri spaces) or more generally, ball–homogeneous spaces, and C-spaces are locally homogeneous or not. We provide some partial positive answers by proving that five–dimensional locally φ–symmetric spaces can be characterized as Sasakian spaces which are ball–homogeneous ...

We study the spinorial Killing equation of supergravity involving a torsion 3-form T as well as a flux 4-form F. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel G2-geometries and on the homogeneous Aloff-Wallach space. The constraint F · Ψ = 0 defines a non empty subfamily of solutions. We investigate the constraint T ·Ψ = 0,...