  Nonrelativistic string theory is described by a sigma model with a
relativistic worldsheet and a nonrelativistic target spacetime geometry, that
is called string Newton-Cartan geometry. In this paper we obtain string
Newton-Cartan geometry as a limit of the Riemannian geometry of General
Relativity with a fluxless two-form field. We then apply the same limit to
relativistic string theory in curved background fields and show that it leads
to nonrelativistic string theory in a string Newton-Cartan geometry coupled to
a Kalb-Ramond and dilaton field background. Finally, we use our limiting
procedure to study the spacetime equations of motion and the T-duality
transformations of nonrelativistic string theory. Our results reproduce the
recent studies of beta-functions and T-duality of nonrelativistic string theory
obtained from the microscopic worldsheet definition of nonrelativistic string
theory.
