  We develop a formalism that allows us to write actions for multiple D-branes
with manifest general covariance. While the matrix coordinates of the D-branes
have a complicated transformation law under coordinate transformations, we find
that these may be promoted to (redundant) matrix fields on the transverse space
with a simple covariant transformation law. Using these fields, we define a
covariant distribution function (a matrix generalization of the delta function
which describes the location of a single brane). The final actions take the
form of an integral over the curved space of a scalar single-trace action built
from the covariant matrix fields, tensors involving the metric, and the
covariant distribution function. For diagonal matrices, the integral localizes
to the positions of the individual branes, giving N copies of the single-brane
action.
