Geometric classifications of k-almost Ricci solitons admitting paracontact metrices

Abstract

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