Derivative corrections to the D-brane Born-Infeld action: non-geodesic embeddings and the Seiberg-Witten map

We dimensionally reduce the four-derivative corrections to the parity-conserving part of the D9-brane effective action involving all orders of the gauge field, to obtain corrections to the actions for the lower-dimensional Dp-branes. These corrections involve the second fundamental form and correspond to a non-geodesic embedding of the Dp-brane into (flat) ten-dimensional space. In addition, we study the transformation of the corrections under the Seiberg-Witten map relating the ordinary and non-commutative theories. A speculative discussion about the higher-order terms in the derivative expansion is also included.