T-duality of a bosonic string in a weakly curved space-time

In this article we consider T-dualization of a $3D$ closed bosonic string that is propagating in space-time metric that has infinitesimal linear dependence on the coordinates $x^{\mu }$. Other fields, Kalb-Ramond and dilaton fields are set to zero. Action with this configuration of fields is not invariant to translations. In order to find T-dual theory we will employ generalization of the Buscher procedure that can be applied to such cases where we have coordinate dependent fields that do not possess translational isometry. Finally, using transformation laws that connect coordinates of starting and T-dual theories, we will be able to examine the geometric structure of T-dual theory.