T-duality in a weakly curved background

We consider the closed string propagating in a weakly curved background which consists of a constant metric and a Kalb–Ramond field with an infinitesimally small coordinate-dependent part. We propose a procedure for constructing the T-dual theory, performing T-duality transformations along the coordinates on which the Kalb–Ramond field depends. The theory obtained is defined in the non-geometric double space, described by the Lagrange multiplier yμ and its T-dual in the flat space y~μ. We apply the proposed T-duality procedure to the T-dual theory and obtain the initial one. We discuss the standard relations between the T-dual theories that imply that the equations of motion and momentum modes of one theory are the Bianchi identities and the winding modes of the other.