Ricci–Yamabe solitons on a class of generalized Sasakian space forms

In this paper, we characterize Ricci–Yamabe solitons and gradient Ricci–Yamabe solitons on 3-dimensional generalized Sasakian space forms with quasi Sasakian metric. Furthermore, we study [Formula: see text]-Ricci–Yamabe solitons and gradient [Formula: see text]-Ricci–Yamabe solitons on 3-dimensional generalized Sasakian space forms with quasi Sasakian metric. Finally, we construct an example to verify a result of our paper.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

∗-Ricci Soliton within the frame-work of Sasakian and (κ,μ)-contact manifold

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere [Formula: see text]. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian [Formula: see text]-contact manifolds.

QUASI-EINSTEIN CONTACT METRIC MANIFOLDS

We consider quasi-Einstein metrics in the framework of contact metric manifolds and prove some rigidity results. First, we show that any quasi-Einstein Sasakian metric is Einstein. Next, we prove that any complete K-contact manifold with quasi-Einstein metric is compact Einstein and Sasakian. To this end, we extend these results for (κ, μ)-spaces.

Examples of compact K-contact manifolds with no Sasakian metric

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.