### 4.2 Stationary toroidal Kaluza–Klein black holes

The four-dimensional vacuum Einstein equations simplify considerably in the stationary and
axisymmetric setting by reducing to a harmonic map into the hyperbolic plane (see Sections 8
and 3.2.3). A similar such reduction in -dimensions works when the isometry group
includes , i.e., besides the stationary vector there exist commuting axial Killing
vectors.
Since the center of has dimension

in the asymptotically flat case the existence of such a group of isometries is only possible for
or . However, one can move beyond the usual asymptotic-flatness and consider
instead -asymptotically-flat spacetimes, in the sense of Section 2.2, with asymptotic ends
, satisfying and , with the isometry group containing
, . Here one takes , with the reference metric of the
form , where is the flat -torus metric. Finally, the action of on
by isometries is assumed in the exterior region to take the form
Such metrics will be referred to as stationary toroidal Kaluza–Klein metrics.