The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have studied solitons on where Ricci tensors are Codazzi type and cyclic parallel. also discussed some curvature conditions admitting vector field torse-forming. shown an example manifold with respect soliton verify our results.

We set the goal to study properties of invariant submanifolds hyperbolic Sasakian manifolds. It is proven that a three-dimensional submanifold manifold totally geodesic if and only it invariant. Also, we discuss ?-Ricci-Bourguignon solitons on Finally, construct non-trivial example five-dimensional validate some our results.

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function...

We examine topological properties of the seven-dimensional positively curved Eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic. A special subfamily of these manifolds also carries a 3-Sasakian metric. Among these we construct a pair of 3-Sasakian spaces which are diffeomorphic to each other, thus giving rise to the first example of a manifold which carri...