Some results on almost Ricci solitons and geodesic vector fields (Q1646652)

An almost Ricci soliton is a Riemannian manifold \((M, g)\) satisfying the conditon \(\mathcal{L}_Vg=2\text{Ric}(g)=2\lambda g\), where \(V\) is a smooth vector field, \(\text{Ric}\) is the Ricci tensor of \(g\), \(\mathcal{L}_V\) is the Lie derivative along \(V\), and \(\lambda\) is a real smooth function on \(M\). If \(\lambda\) is constant, it reduces to a Ricci soliton. The author proves that a compact almost Ricci soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. It is also shown that an almost Ricci soliton reduces to a Ricci soliton if and only if the associated vector field is geodesic. This means that \(\square X=0\), where \(\square X\) is a vector field with components \(-(g^{jk}\nabla_j\nabla _kX^i+R_j^iX^j)\) in a local coordinate system \(x^j\). Finally, he proves that a contact metric manifold is \(K\)-contact if and only if its Reeb vector field is geodesic.