Riemann Soliton within the framework of contact geometry

Abstract

In this paper, we study contact metric manifold whose metric is a Riemann soliton. First, we consider Riemann soliton (g; V ) with V as  contact vector eld on a Sasakian manifold (M; g) and in this case we prove that M is either of constant curvature +1 (and V is Killing) or D-homothetically xed -Einstein manifold (and V leaves the structure tensor φ invariant). Next, we prove that if a compact K-contact manifold whose metric g is a gradient almost Riemann soliton, then it is Sasakian and isometric to a unit sphere S²ⁿ⁺¹. Further, we study H-contact  manifold admitting a Riemann soliton (g; V ) where V is pointwise collinear with .

Key words: Contact metric manifold, Riemann soliton, gradient almost Riemann soliton.