Contact GRA Solitons and Applications to General Relativity

This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete K-contact or Sasakian manifold endowed with a closed GRA soliton satisfying \(4c_1c_2 \ne 1\) is compact Einstein with scalar curvature \(2n(2n+1)\). As for the gradient case, it exhibits an isometry to the unit sphere \({\mathbb {S}}^{2n+1}\). Subsequently, we identify a few adequate conditions under which a non-trivial complete K-contact manifold with a GRA soliton is trivial (\(\eta \)-Einstein). Following that, we establish certain results on H-contact and complete contact manifolds. We also demonstrate that a non-Sasakian \((k,\mu )\)-contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle \({\mathbb {R}}^{n+1} \times {\mathbb {S}}^n(4)\), provided \(4c_1c_2 (1-2n)\ne 1\) and \(c_2\ne 0\). Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime.