We analyse open string correlators in non-constant background fields, including the metric g, the antisymmetric B-field, and the gauge field A. Working with a derivative expansion for the background fields, but exact in their constant parts, we obtain a tachyonic on-shell condition for the inserted functions and extract the kinetic term for the tachyon action. The 3-point correlator yields a non-commutative tachyon potential. We also find a remarkable feature of the differential structure on the D-brane: Although the boundary metric G plays an essential role in the action, the natural connection on the D-brane is the same as in closed string theory, i.e. it is compatible with the bulk metric and has torsion H = dB. This means, in particular, that the parallel transport on the brane is independent of the gauge field A.