$\parbox{0.45\textwidth}{\begin{flushleft}\vspace{-\baselineskip} {\L... ...tet}\\ \textbf{Matematiska Institutionen}\\ Thomas Erlandsson \end{flushleft}}$

DIFFERENTIAL GEOMETRY MN1 FALL 1999

PROBLEM 12 PAGE 3

The Poincaré Disc Model of H²

The surface $\,H^2\,$ in the Poincaré disc model is the set

$$U=\{(u,v) \in R^2\vert u^2+ v^2 < 4\},$$

with the Riemann metric

$$\,ds^2=\left(1-\frac{u^2 + v^2}{4}\right)^{-2} (du^2+dv^2).$$

Using Gauss' equation we find that the surface has constant Gauss curvature $\,K=-1.$ The geodesics in the disc model correspond to the circles orthogonal to the boundary of the disc and the diameters. This is most easily seen by showing that the map

$$w=\frac{z+2i}{iz+2}$$

is an isometry of the disc model onto the half plane model.

d)

Calculate the arc length $\,r\,$ of the geodesic $\,c(t)=(t\cos \vartheta, t\sin \vartheta),\,\,0 \le t < 2\,$ beginning at origo.

$\quad$ (Result: $r=\ln\frac{2+t}{2-t}=2 \frac{1}{2}\ln\frac{1+t/2}{1-t/2}= 2\tanh^{-1}(t/2)$)

e)

Show that in geodesic polar coordinates

$$ds^2 = dr^2+\sinh ^{2}(r)\, d\theta^2.$$

Hint: From the problem above follows that $\,t = 2\tanh (r/2).$
Use $\,u=t\cos\theta, v=t\sin\theta\,$ and the given metric.

To Problem 12 page 1

To Problem 12 page 2

Tillbaka till Differentialgeometri MN1