T-duality of the Riemann curvature corrections to supergravity

We examine the known Riemann curvature corrections to the supergravity action at order $\alpha'^3$ under the T-duality transformations. Using the compatibility of the action with the linear T-duality and with the S-matrix calculations as guiding principles, we have found the couplings of four $B$-field strengths and the couplings of two Riemann curvatures and two $B$-field strengths at order $\alpha'^3$. Using the S-matrix calculations we have also found new dilaton couplings in the string frame at this order.