  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.
