  In this paper we study the action for N D0-branes in a curved background. In
particular, we focus on the meaning of space-time diffeomorphism invariance.
For a single D-brane, diffeomorphism invariance acts in a naive way on the
world-volume fields, but for multiple D-branes, the meaning of diffeomorphism
invariance is much more obscure. The problem goes beyond the determination of
an ordering of the U(N)-valued fields, because one can show that there is no
lift of ordinary diffeomorphisms to matrix-valued diffeomorphisms. On the other
hand, the action can presumably be constructed from perturbative string theory
calculations. Based on the general characteristics of such calculations we
determine a set of constraints on the action for N D0-branes, that ensure
space-time covariance. These constraints can be solved order by order, but they
are insufficient to determine the action completely. All solutions to the
constraints obey the axioms of D-geometry. Moreover the action must contain new
terms. This exhibits clearly that the answer is more than a suitable ordering
of the action of a single D0 brane.
