6.3 Wave-map structure
It is of interest to note that Equations (16) – (17) are of wave-map type. The domain is given by
with the metric
and the target is
with the metric
Equations (16) – (17) are the wave-map equations for a map from
to
, which is
independent of the
-coordinate. In other words, they coincide with the Euler–Lagrange equations
corresponding to the action
6.3.1 Representations of hyperbolic space
It is of interest to note that
is isometric to hyperbolic space. In fact,
gives an isometry from
to the upper half plane model of hyperbolic space,
, where
and
For certain considerations, the upper half plane is an inappropriate model of hyperbolic space. The reason is
that the upper half plane has a preferred boundary point, namely infinity; see Section 9.3. The disc model,
given by the Riemannian metric
on the open unit disc
does not have preferred boundary points, and therefore is sometimes preferable.
Using complex notation,
yields an isometry from the upper half plane to the disc model. By composing
with
, we
obtain an isometry from
to
. The expression is given by