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.

Keywords