Higher-derivative couplings and torsional Riemann curvature

Using the most general higher-derivative field redefinition for the closed spacetime manifolds, we show that the tree-level couplings of the metric, $B$-field and dilaton at orders $\alpha'^2$ and $\alpha'^3$ that have been recently found by the T-duality, can be written in a particular scheme in terms of the torsional Riemann curvature ${\cal R}$ and the torsion tensor $H$. The couplings at order $\alpha'^2$ have structures ${\cal R}^3, H^2 {\cal R}^2$, $H^6$, and the couplings at order $\alpha'^3$ have only structures ${\cal R}^4$, $H^2{\cal R}^3$. Replacing ${\cal R}$ with the ordinary Riemann curvature, the couplings in the structure $H^2{\cal R}^3$ reproduce the couplings found in the literature by the S-matrix method.