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\centerline{MATH 392 -- Geometry Through History}
\centerline{Discussion -- An interesting abstract surface}
\centerline{April 22 and 25, 2016}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
As we have seen, it's possible to define Riemannian metrics, Gaussian curvature, and geodesics
on abstract surfaces once we have a good analog of the tangent space at each point.  
(But unlike the situation of a surface in $\mathbb{R}^3$, we had to cook up a pretty
non-intuitive way to define the elements the elements of the tangent space.  They
are constructed as directional derivative operators on functions on the surface
rather than as ``physical'' vectors.  But everything works satisfactorily!)
We are going to spend the rest of the time before the final project presentations studying
a particularly interesting and historically significant example.  
Namely, let $\mathbb{H}$ as a set be
$$\mathbb{H} = \{(x,y) \in \mathbb{R}^2\mid y > 0\}$$
the ``upper half-plane.''  We will think of this as an abstract surface \emph{different from
the plane or a part of the ordinary plane} by introducing the Riemannian metric with 
$E = G = \frac{1}{y^2}$ and $F = 0$, or
\begin{equation}
\label{PoincareMetric}
ds^2 = \frac{dx^2 + dy^2}{y^2}.
\end{equation}
We want to study the geometry on this abstract surface in detail and understand
how it compares with other geometries we have studied in the course.
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\noindent
{\it Questions}
\vskip 10pt
\begin{enumerate}
\item[(A)] To ``warm up'' and see what (\ref{PoincareMetric}) does to make the geometry
of this abstract surface significantly different from the usual geometry of the upper half-plane in $\mathbb{R}^2$, for each $u \in (0,1)$
consider the curve $\alpha_u(t) = (0,1) + t(0,-u) = (0,1 - tu)$ for $t \in [0,1]$.  
Using the metric from (\ref{PoincareMetric}), compute the length of $\alpha_u(t)$ 
as a function of $u$.  What happens in the limit as $u \to 1$, so $\alpha_u(1) \to (0,0)$?
\item[(B)] Compute the Christoffel symbols and Gaussian curvature for this abstract 
surface.  The result for the curvature should be surprisingly simple(!)
\item[(C)] Show that for any constant $a$, the curve $\beta(t) = (a,e^t)$ is a unit
speed geodesic on this abstract surface.
\item[(D)] Show that if the constants $a,b,c,d$ satisfy $ad - bc = 1$, then the 
following curve is a geodesic on this abstract surface\footnote{(continues at bottom of page 2) This is a pretty messy calculation by hand (although it's definitely do-able).  If
you want, you can use Maple to show that the differential equations for geodesics
are satisfied.  The ``clean way'' to prove this is to show that this curve is actually
the image of the geodesic from part (C) under a certain transformation of $\mathbb{H}$
that preserves the metric (\ref{PoincareMetric}).  The cleanest way to describe
that mapping is by writing $(x,y)$ as the complex number $z = x + iy$.  Then the mapping
involved is the so-called \emph{fractional linear} mapping $z \mapsto \frac{az + b}{cz + d}$.}:
\begin{equation}
\label{geodesic}
\gamma(t) = \left(\frac{ac e^{2t} + bd}{c^2 e^{2t} + d^2}, \frac{e^t}{c^2 e^{2t} + d^2}\right)
\end{equation}
\item[(E)]  Show that $\gamma(t)$ from part (D) is an arc of a circle in the plane whose diameter
is a segment along the $x$-axis.  As $t \to \pm \infty$, show that the tangent vector to 
$\gamma(t)$ becomes vertical, so although $\gamma(t)$ never reaches the $x$-axis, it 
approaches the $x$-axis along a perpendicular direction as $t \to \pm\infty$.  
\item[(F)]  Show that given any two points $P,Q$ in $\mathbb{H}$, either 
$P,Q$ have the same $x$-coordinate so they are both on one of the geodesics from (C), or
else there is a curve of the form (\ref{geodesic}) containing both $P,Q$  (Hint:  use the 
description given in part (E)!)  Explain why the geodesic segment from $P$ to $Q$ can
always be extended indefinitely in both directions.  
\item[(G)]  A \emph{geodesic circle} in $\mathbb{H}$ with center at $P$ and radius
$r > 0$ should be the curve obtained by moving out from $P$ a distance $r$ along the 
geodesics through $P$ in all possible directions.  The distance should be measured
as arc-length along the geodesic.  Argue that geodesic circles exist with all centers
and all radii (all values of $r > 0$).  
\item[(H)]  Right angles are defined using the metric to compute angles between vectors.
In the tangent space, these are always right angles in the usual sense, so they
are all the same.  (Nothing to prove here, just notice the pattern.)
\item[(I)]  Given a geodesic $\ell$ and a point $P$ not on the geodesic, how many
geodesics are there through $P$ that do not meet $\ell$?
\item[(J)]  What's the geometry on this abstract surface $\mathbb{H}$ with 
the metric from (\ref{PoincareMetric})?  Where have we seen it before?
\end{enumerate}
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\noindent
{\it Assignment}
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Group writeups due no later than Friday, May 6.  (This will be the final assignment for 
the course apart from the final project presentations and papers.)
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