An asymptotically flat manifold is a Riemannian manifold with an end diffeomorphic to the complement of a ball in 
on which the metric approaches the Euclidean
metric. A differentiated decay condition commonly used for initial data is
 |
where 
is the extrinsic curvature tensor of the initial data set, 
, and 
bounds derivatives of order 
by a constant times 
. The decay exponent and number of derivatives are part
of the hypotheses of any mass theorem.
A manifold can have several such ends, each with its own ADM energy and momentum.
