2D Ricci flat gradient solitons arising from remarkable models in physics (Q2865248)

An \(n\)-dimensional pseudo-Riemannian manifold \((M,g)\) is a gradient Ricci soliton if there exists a smooth real function \(f\) on the manifold satisfying \(R_{ij} + (\nabla^2_g f)_{ij} = \rho g_{ij}\) for some real constant \(\rho\), where \((\nabla^2_g f)_{ij}\) are the components of the pseudo-Riemannian Hessian of \(f\). If this Hessian is non-degenerate, the author gives some conditions for the associated Levi-Civita connections of \(g\) to coincide with the Levi-Civita connection of this Hessian. As an application in 2-dimensions, some classes of Ricci flat gradient solitons are obtained for some particular cases of metrics \(g\), which arise from physical models and have constant sectional curvature.