### 2.2 Kaluza–Klein asymptotic flatness

There exists a generalization of the notion of asymptotic flatness, which is relevant to both four- and
higher-dimensional gravitation. We shall say that is a Kaluza–Klein asymptotic end if is
diffeomorphic to , where is a closed coordinate ball of radius and is a
compact manifold of dimension ; a spacetime containing such an end is said to have
asymptotically-large dimensions. Let be a fixed Riemaniann metric on , and let , where
is the Euclidean metric on . A spacetime containing such an end will be said to be
Kaluza–Klein asymptotically flat, or -asymptotically flat if, for some , the metric induced
by on and the extrinsic curvature tensor of , satisfy the fall-off conditions
where, in this context, is the radius in and we write if satisfies
with the Levi-Civita connection of .