Covariant Poisson's equation in torsional Newton-Cartan gravity

We derive the covariant Poisson's equation of (d + 1)-dimensional Newton-Cartan gravity with (twistless) torsion by applying the non-relativistic conformal method' introduced in [1]. We apply this method on-shell to a Schrodinger field theory on the curved Newton-Hooke background. The covariance of the field equation in the presence of the non-relativistic cosmological constant, entails fixing all coefficients in the covariant Poisson's equation for (twistless) torsional Newton-Cartan gravity. We further derive Ehlers conditions and an equation associated to the torsion in this method.