Narrow Results

The main purpose of the paper is to study an almost gradient Ricci–Bourguignon soliton (RB soliton) within the framework of $K$-contact manifolds and $(\kappa ,\psi )$-contact manifolds. First, we prove that if complete $K$-contact manifold endows a gradient RB soliton, then the manifold is compact Sasakian and isometric to unit sphere $S^{2n+1}$. Next, we show that if a complete contact metric satisfies an almost RB soliton with a non-zero potential vector field is collinear with the Reeb vector field $\xi$ and the Reeb vector field $\xi$ acting as an eigenvector of the Ricci operator, then it is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $\xi$. Lastly, we prove that if the metric of a non-Sasakian $(\kappa ,\psi )$-contact manifold is an almost gradient RB soliton, then it is flat in dimension 3 and in higher dimensions it is locally isometric to $E^{n+1}\times S^n(4)$.

We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to Eⁿ⁺¹ × Sⁿ(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ² and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.