6.3 Wave-map structure

It is of interest to note that Equations (16View Equation) – (17View Equation) are of wave-map type. The domain is given by 2 ℝ × T with the metric
g = − e− 2τdτ 2 + d 𝜃2 + e− 2τd χ2 0

and the target is 2 ℝ with the metric

2 2P 2 gR = dP + e dQ . (20 )
Equations (16View Equation) – (17View Equation) are the wave-map equations for a map from (ℝ × T2,g0) to (ℝ2, gR), which is independent of the χ-coordinate. In other words, they coincide with the Euler–Lagrange equations corresponding to the action
∫ ∫ 2 2P 2 −2τ 2 2P 2 [− Pτ − e Q τ + e (P𝜃 + e Q𝜃)]d𝜃dτ.

6.3.1 Representations of hyperbolic space

It is of interest to note that 2 (ℝ ,gR ) is isometric to hyperbolic space. In fact,

ϕRH (Q, P ) = (Q,e− P) (21 )
gives an isometry from 2 (ℝ ,gR) to the upper half plane model of hyperbolic space, (H, gH), where 2 H = {(x, y) ∈ ℝ : y > 0} and
2 2 gH = dx--+-dy-. (22 ) y2
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
4(dx2 + dy2 ) gD = --------------, (23 ) (1 − x2 − y2)2
on the open unit disc D does not have preferred boundary points, and therefore is sometimes preferable. Using complex notation,
z − i ϕHD = ----- z + i

yields an isometry from the upper half plane to the disc model. By composing ϕHD with ϕRH, we obtain an isometry from (ℝ2,gR ) to (D, gD ). The expression is given by

− P ϕRD (Q, P ) = Q-+--i(e---−--1). (24 ) Q + i(e− P + 1)

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