Geometric classifications of k-almost Ricci solitons admitting paracontact metrices

Abstract

The prime objective of the approach is to give geometric classifications of kk-almost Ricci solitons associated with paracontact manifolds. Let M2n+1(φ,ξ,η,g){M}^{2n+1}\left(\varphi ,\xi ,\eta ,g) be a paracontact metric manifold, and if a KK-paracontact metric gg represents a kk-almost Ricci soliton (g,V,k,λ)\left(g,V,k,\lambda ) and the potential vector field VV is Jacobi field along the Reeb vector field ξ\xi , then either k=λ−2nk=\lambda -2n, or gg is a kk-Ricci soliton. Next, we consider KK-paracontact manifold as a kk-almost Ricci soliton with the potential vector field VV is infinitesimal paracontact transformation or collinear with ξ\xi . We have proved that if a paracontact metric as a kk-almost Ricci soliton associated with the non-zero potential vector field VV is collinear with ξ\xi and the Ricci operator QQ commutes with paracontact structure φ\varphi , then it is Einstein of constant scalar curvature equals to −2n(2n+1)-2n\left(2n+1). Finally, we have deduced that a para-Sasakian manifold admitting a gradient kk-almost Ricci soliton is Einstein of constant scalar curvature equals to −2n(2n+1)-2n\left(2n+1).

Keywords

• k-almost ricci solitons
• ricci soliton
• einstein manifold
• paracontact metric manifold
• infinitesimal paracontact transformation
• 53c25
• 53c21
• 53d15