Poincare's UHP versus disk models of the hyperbolic plane
Careful what you ask for, I might toss a small book at you!
You are correct that the Moebius transformation
w \mapsto \frac{w+i}{w-i}
takes 0, \, i, \, \infty to -1, \, 0, \, 1 and thus maps the UHP
conformally to the interior of the unit disk. If you express this as a real transformation, we can regard it as a
coordinate transformation from
Poincare's upper half plane model of the hyperbolic plane, which in the
UHP chart has the line element
<br />
ds^2 = \frac{dx^2+dy^2}{y^2}, \; -\infty < x < \infty, \; 0 < y < \infty
to
Poincare's unit disk model of the hyperbolic plane, which in the
polar stereographic chart has the line element
<br />
ds^2 = \frac{4 \, \left( d\rho^2 + \rho^2 \, d\phi^2 \right)}{1-\rho^2}, \;<br />
0 < \rho < 1, \; -\pi < \phi < \pi<br />
This chart is readily transformed to the more familar
cartesian stereographic chart with the line element
<br />
ds^2 = \frac{4 \, \left( d\overline{x}^2+d\overline{y}^2 \right)}<br />
{\left( 1-\overline{x}^2-\overline{y}^2 \right)^2}, \;<br />
\overline{x}^2+\overline{y}^2 < 1<br />
which can indeed be obtained by stereographic projection of the hyperboloid of one sheet considered as the variety -x_1^2 + x_2^2 + x_3^2 = -1 in E^{1,2} from the point (x_1,x_2,x_3)=(-1,0,0) to the plane x_1 = 0.
Poincare himself recalled, in a famous passage in one of his popular books written for a general audience, that he noticed these facts at the exact moment in which he set his foot down to get onto a bus!
To see this, consider the inverse transformation
<br />
z \mapsto \frac{i \, (1-z)}{1+z}<br />
Writing z = \rho \, \exp(i \, \phi), the transformation becomes
<br />
x = \frac{2 \rho \, \sin(\phi)}{\rho^2 + 2 \rho \, \cos(\phi) + 1}, \;<br />
y = \frac{1-\rho^2}{\rho^2 + 2 \rho \, \cos(\phi) + 1}<br />
with inverse transformation
<br />
\rho = \sqrt{ \frac{x^2+(y-1)^2}{x^2+(y+1)^2}}, \;<br />
\phi = \arctan \frac{2 x}{1-x^2-y^2}<br />
Plugging into the first line element above, we obtain the second.
Let's study H^2 in the UHP chart.
You can readily solve the Killing equations to find three generators of the
Lie algebra of Killing vectors, namely
<br />
\partial_x, \; <br />
x \, \partial_x + y \, \partial_y, \; <br />
\frac{x^2-y^2}{2} \, \partial_x + x y \, \partial_y<br />
which I hope you recognize as three
infinitesimal Moebius transformations!
To obtain the uniparameteric subgroups of transformations corrresponding to each generator, we proceed as follows:
To find the integral curves of the vector field \partial_x, we solve
<br />
\dot{x} = 1, \; \dot{y} = 0, \; x(0)=x_0, \; y(0) = y_0<br />
which gives x(s) = x_0 + s, \; y(s) = y_0, or better yet, (x,y) \mapsto (x + \lambda, y), i.e.
horizontal translation .
To find the integral curves of the vector field x \, \partial_x + y \, \partial_y, we solve
<br />
\dot{x} = x, \; \dot{y} = y, \; x(0)=x_0, \; y(0) = y_0<br />
which gives x(s) = x_0 \, \exp(s), \; y(s) = y_0 \, \exp(s) ("additive" group parameterized by s), or better yet, (x,y) \mapsto (x \, \lambda, y \, \lambda) ("multiplicative" group parameterized by \lambda > 0). That is, dilation from the origin in euclidean terms, or "vertical"
translation in hyperbolic terms.
Exercise: find the integral curves of the vector field \frac{x^2-y^2}{2} \, \partial_x + x y \, \partial_y; the resulting one-parameter group of transformations should consist of the
hyperbolic rotations about the point (x,y)=(0,1), with orbits which are coordinate circles with various centers which in hyperbolic terms are concentric circles around this point.
Thus, we have a three dimensional Lie group of self-isometries of the hyperbolic plane, which is fully analogous to the three dimensional Lie group of self-isometries of the euclidean plane.
The geodesic equations are readily obtained from the
geodesic Lagrangian; the result is
<br />
\ddot{x} - \frac{2 \dot{x} \dot{y}}{y} = 0, \; <br />
\ddot{y} + \frac{\dot{x}^2-\dot{y}^2}{y} = 0<br />
From this we readily obtain the first integrals
<br />
\dot{x} = A \, y^2, \;<br />
\dot{y} = y \, \sqrt{(1- A^2 \, y^2}<br />
From this we obtain
<br />
\frac{dx}{dy} = \frac{\dot{x}}{\dot{y}} = \frac{A y}{\sqrt{1-A^2 y^2}}<br />
which gives (x-x_0-\sqrt{y_m-y_0^2})^2 + y^2 = y_m^2. This is evidently, in euclidean terms, a semicircular arc passing through (x,y) = (x_0, y_0), orthogonal to y=0, and with maximal height y=y_m.
From our knowledge of conformal mappings in general and Moebius transformations in particular, we can immediately infer that in the unit disk model, the geodesics appear (in euclidean terms) as semicircular arcs orthogonal to the unit circle (the image of the real line under our diffeomorphic mapping giving the change of coordinate charts as above). We could also confirm this directly by studying the geodesics in the polar stereographic chart. Readers should draw a sketch at this point to verify that given a line L (geodesic) in H^2 and a point P off L, there are infinitely many lines through P which do not intersect L!
PF readers will no doubt be familiar with formulas from analytic geometry giving the equation of the line L through two given points P1, P2, aka the
straightedge equation, and also the equation of a circle of radius A with center P, aka the
compass equation.
Exercise: find a formula giving the equation of the hyperbolic geodesic through two points P1, P2 in the unit disk. Find a formula giving the equation of the circle of hyperbolic radius A and hyperbolic center P.
Once one has these in hand, note that most euclidean constructions with straightedge and compass do not depend upon the parallel postulate, so we can simply substitute our hyperbolic versions. In this way we can confirm that if we assume that euclidean geometry is self-consistent, we must conclude that hyperbolic geometry is also self-consistent. In this way Gauss and independently Lobachevski and Janos Bolyai built up the synthetic theory of hyperbolic geometry.
There is no need to use analytic formulas here; what GLB actually did was to use euclidean constructions with straightedge and compass to model the required hyperbolic straighedge and compass. See Chaim Goodman-Strauss, "Compass and Straightedge in the Poincare Disk",
American Mathematical Monthly 108 (2001): 38-49, for a detailed discussion.
Actually, Poincare's flash of inspiration covered
much more ground than we can cover here; he also saw how
doubly periodic functions encountered in complex analysis (these arise naturally for example in solving various ODEs) are connected with
tilings of the hyperbolic plane and with their symmetry groups. See for example Jones and Silverman,
Complex Functions for doubly periodic groups and see Armstrong,
Groups and Symmetry for an elementary enumeration of the symmetry groups.
I feel that we ought to have a sticky listing some examples of structure at various levels in manifold theory and Riemannian (or Lorentzian) geometry. As an example of global structure, we see that H^2 is aptly named, for it is indeed diffeomorphic (and thus homeomorphic) to the euclidean plane, but because {R\left[H^2\right]}_{1212} = -1[/tex] while {R\left[E^2\right]}_{1212} = 0, it is certainly not <i>isometric</i> to the euclidean plane! Furthermore, from considering "ideal endpoints" (on the unit circle) of geodesics in the disk model, you can see that the space of lines (geodesics) is diffeomorphic to a Moebius band, i.e. real projective plane with a disk removed.<br />
<br />
There are many other charts on H^2 which are often encountered, including the <i>radial chart</i><br />
<br />
ds^2 = \frac{dr^2}{1+r^2} + r^2 \, d\phi^2, \; <br />
0 &lt; r &lt; \infty, \; -\pi &lt; \phi &lt; \pi<br /><br />
the <i>cylindrical conformal chart</i> (analog of Mercator chart, in which <i>loxodromes</i> appear as coordinate lines, which gives an angle-preserving but not area-preserving euclidean representation)<br />
<br />
ds^2 = \frac{d\zeta^2 + d\phi^2}{\sinh(\zeta)^2}, \; <br />
0 &lt; \zeta &lt; \infty, \; -\pi &lt; \phi &lt; \pi<br /><br />
the <i>cylindrical axial projection</i> chart (area preserving)<br />
<br />
ds^2 = \frac{dz^2}{z^2-1} + \left( z^2-1 \right) \, d\phi^2, \;<br />
-1 &lt; z &lt; 1, \; -\pi &lt; \phi &lt; \pi<br /><br />
the <i>radial trig chart</i><br />
<br />
ds^2 = d\theta^2 + \sinh(\theta)^2 \, d\phi^2, \;<br />
0 &lt; \theta &lt; \pi, \; -\pi &lt; \phi &lt; \pi<br /><br />
and the <i>planar central projection</i> chart<br />
<br />
ds^2 = \frac{dv^2}{\left( 1-v^2 \right)^2} + \frac{v^2}{1-v^2} \, d\phi^2 , \;<br />
0 &lt; v &lt; 1, \; -\pi &lt; \phi &lt; \pi<br /><br />
which is geodesic-preserving; the geodesics appear as coordinate line segments in the unit disk, which gives <i>Klein's unit disk model</i> of H^2.<br />
<br />
Exercise: find explicit coordinate transformations between these, and determine which properties each chart has: angle-preserving, area-preserving, or geodesic-preserving? Some of these are best considered to "live" in part of the plane, and some, in part of a cylinder. Which are which? (Some hints provided above.)<br />
<br />
Exercise: study Flanders, <i>Differential Forms and their Physical Applications</i>. For each chart, read off a natural coframe field. If neither unit vector gives a geodesic vector field, rotate by a suitable amount at each point to obtain a frame field which is geodesic. Express the Laplace-Beltrami operator in terms of each chart. Express the three Killing vectors in terms of each chart.<br />
<br />
Exercise: study Olver, <i>Applications of Lie Groups to Differential Equations</i>. Explain why the heat equation in the hyperbolic plane becomes (in the UHP chart)<br />
<br />
\frac{u_t}{y^2} = u_{xx} + u_{yy}<br /><br />
Determine its point symmetry group and find the fundamental solution using the methods of Lie.<br />
<br />
<blockquote data-attributes="" data-quote="WWGD" data-source="post: 1515460"
class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch">
<div class="bbCodeBlock-title">
WWGD said:
</div>
<div class="bbCodeBlock-content">
<div class="bbCodeBlock-expandContent js-expandContent ">
do diffeomorphisms preserve geodesics?
</div>
</div>
</blockquote><br />
I suggested above considering diffeomorphisms as coordinate transformations. From this point of view, the fact that our transformation arises from a conformal mapping in the sense of complex variables ensures that since our first chart was "conformal", our second chart (obtained by applying this mapping as a coordinate transformation) must also be "conformal", and it is.<br />
<br />
As suggested above, we can compare any coordinate chart on any Riemannian two-manifold with a locally euclidean model (above I used some cylindrical and some planar models), and we can decide whether or not our chart gives a euclidean representation of the intrinsic geometry on our Riemannian two-manifold which preserves angles, geodesics, or area. Gauss himself showed that no representation can preserve all three if the Gaussian curvature is nonzero. But as we saw above we can find representations which preserve either angles or geodesics or area!<br />
<br />
Exercise: for the sphere S^2, find analogous charts and work out exercises analogous to the ones above. The same, for H^{1,1}.
