  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 $A$.
