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
.