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\centerline{MATH 392 -- Geometry Through History}
\centerline{Final Class Before Project Presentations}
\centerline{April 29, 2016}
\vskip 10pt
As you have now seen, the abstract surface $\mathbb{H}$ consisting of the 
upper half-plane with the Riemannian metric 
$$ds^2 = \frac{dx^2 + dy^2}{y^2}$$
is what is called a ``model'' of the geometry of the hyperbolic plane
we studied earlier in the semester(!) In other words, it is a surface constructed
using the tools from differential geometry whose geodesics have the same properties
as the lines in hyperbolic plane.  The usual name for this surface is the {\it Poincar\'{e}
upper half-plane model}.  The existence of examples like this one in effect
provides additional evidence for the validity of hyperbolic geometry.  

As we said earlier in the semester, non-Euclidean geometries first appeared
in published form in work of Lobachevsky (1830) and Bolyai (1832).  Their
work was to a great extent anticipated by unpublished work of Gauss done in the
early 1800's.  As their papers became better known and accepted by other
mathematicians, there were still (at least in some quarters) what I will 
call ``lingering doubts'' about whether hyperbolic geometry was really a
consistent mathematical system.  Bolyai and Lobachevsky certainly worked under the 
understanding that their non-Euclidean geometries were just as consistent as
the Euclidean geometry with Postulate V.  However, they did not provide any
\emph{proofs} of such statements; they merely developed all the properties of the 
hyperbolic plane and exhibited all the ways hyperbolic geometry differed
from Euclidean geometry.  

The first mathematician who really addressed the question of proving
consistency was the Italian geometer 
Eugenio Beltrami (1835-1900).  What he did was to use the differential
geometry of surfaces developed by Gauss and others to produce an explicit
surface whose intrinsic geometry would be the same as (a portion of) the 
hyperbolic plane.  This was even a {\it surface
of revolution}, obtained by rotating the curve called the unit speed
curve called the {\it tractrix}
$$\alpha(s) = \left(e^{-s}, \ln(e^s + \sqrt{e^{2s} - 1}) - \frac{\sqrt{e^{2s} - 1}}{e^s}\right)$$
about the $x$-axis.  It is a nice (but somewhat involved) exercise to verify
that the resulting surface, often called the \emph{pseudosphere} (see Figure 1)
has constant Gaussian curvature $K = -1$ (the same as our abstract surface $\mathbb{H}$).  
Beltrami then studied the geodesics on that surface and showed that geodesic
triangles have the form we saw in hyperbolic geometry (the angle sums are
less than $180^\circ$, etc.)  

\begin{figure}
\begin{centering}
\includegraphics[width=3truein,height=3truein]{Pseudosphere.jpeg}
\caption{Plaster model of the pseudosphere}
\end{centering}
\end{figure}

Now the key point here is that this surface with its non-Euclidean intrinsic
geometry has been constructed {\it inside of the familiar Euclidean} $\mathbb{R}^3$.
If there were any contradiction involved in its properties, they would also
be contradictions regarding properties of an ordinary Euclidean surface of 
revolution.  With this insight, what Beltrami showed is often paraphrased by saying that
hyperbolic geometry is ``just as consistent as Euclidean geometry.''  Technically, 
this is known as a {\it relative consistency proof}.  Stated more formally, what
he proved is that {\it Euclidean geometry is consistent if and only if hyperbolic geometry
is consistent}.

However, there is a deficiency of the pseudosphere model of hyperbolic geometry 
in that the pseudosphere does not represent \emph{all of the hyperbolic plane}. 
Hence it's somewhat awkward to say how Postulate II is supposed to hold since 
geodesics cannot be extended indefinitely as they tend toward the points on the 
boundary circle at the bottom of the pseudosphere.  
Beltrami also addressed this issue and developed another model of hyperbolic
geometry.  This model was also studied by the German mathematician Felix Klein (1849-1925)
and is shown in Figure 2.  It is more commonly known as the {\it Klein model} for this
reason, but Beltrami really deserves some of the credit too.  In this model, 
\begin{itemize}
\item The points are the points (strictly) inside the unit disk in $\mathbb{R}^2$, 
and 
\item the geodesics are chords of the boundary circle.
\item The boundary circle is not included, though, so the geodesics are \emph{open} line
segments.  
\item Distances and angles are
defined in terms of a ``non-standard'' Riemannian metric.  
\end{itemize}
As on our example surface
$\mathbb{H}$, this means that the lengths of curves in this model differ from
the lengths of the corresponding Euclidean curves.  {\it Unlike} the case in our 
example surface $\mathbb{H}$, angle measurements are \emph{also different}.  This
means, for instance, that ``right angles'' in this model do not look like Euclidean
right angles.  The fact that they do in our model $\mathbb{H}$ is a special 
property known as \emph{conformality}.  

\begin{figure}
\begin{centering}
\includegraphics[width=3truein,height=3truein]{KleinModel.png}
\caption{The Beltrami-Klein model of the hyperbolic plane}
\end{centering}
\end{figure}

The model we studied in our discussion was developed by a different mathematician
in what started out as a very different context.  One of the great anecdotes about 
mathematical inspiration comes from the French mathematician
Henri Poincar\'e (1854 - 1912).  At one point, Poincar\'e was
deeply involved in trying to solve a problem in complex analysis
regarding what he called ``Fuchsian functions''--functions defined on the
unit disc in the complex plane:
$$\Delta = \{z \in \mathbb{C} : |z| < 1\}$$
that would satisfy particular types of transformation rules.  

Poincar\'e says:
\vskip 10pt
\emph{Just at this time I left Caen, where I was then living, to go on a geological
excursion under the auspices of the school of mines. The changes of travel
made me forget my mathematical work.  Having reached Coutances, we
entered an omnibus to go some place or other. At the moment when I put
my foot on the step the idea came to me, without anything in my former
thoughts seeming to have paved the way for it, that the transformations I
had used to define the Fuchsian functions were identical with those of non-
Euclidean geometry. I did not verify the idea; I should not have had time,
as, upon taking my seat in the omnibus, I went on with a conversation
already commenced, but I felt a perfect certainty. On my return to Caen,
for conscience' sake I verified the result at my leisure.}
\vskip 10pt
In the \emph{Poincar\'e disc model} 
shown in Figure 3\footnote{I believe Beltrami (as well as Riemann) also 
defined something equivalent to this version, and 
well before Poincar\'e did.  Why isn't it named after one of them?  The best answer
is probably that mathematicians are generally very bad historians as you might guess 
already from the Beltrami-Klein case discussed before(!)  Things tend
to get named after more famous mathematicians or after the place most people learn about them, 
not after the first people to develop them. Sometimes also connections are overlooked or
not appreciated if constructions are written down in different ways.} the points are again the points
in the interior of the unit circle.  But now geodesics are arcs of circles
meeting the boundary circle at right angles.  In the figure above, we
see infinitely many such geodesics through a point not lying on the solid
blue geodesic.  The parallels are the two circles meeting the solid blue
curve at points on the boundary circle.  (Those points are not points in
the hyperbolic plane though; we would think of them as ``points at infinity''
and the parallels are merely asymptotic in the hyperbolic plane itself.)

\begin{figure}
\begin{centering}
\includegraphics[width=3truein,height=3truein]{PoincareModel.png}
\caption{The Poincar\'e model of the hyperbolic plane}
\end{centering}
\end{figure}


\vskip 10pt
As a domain for 
defining functions of a complex variable, the upper half plane we studied in the final
small group discussion days is actually equivalent to this disk, via a mapping
that can be described in complex coordinates by 
$$\phi : z \mapsto \frac{z - i}{z + i}$$
(This maps $i \leftrightarrow (0,1)$ to $0$, the $x$-axis to the unit circle
and the upper half-plane to the interior of the unit circle.)
As a result he saw how to create \emph{analytical models} of hyperbolic
geometry using the interior of the unit disc and the upper half-plane
in the complex plane, and do all of the hyperbolic constructions we did in terms
of complex analysis(!) 

So we have reached the end of the part of our journey where I serve as guide.
I hope you have found it an interesting trip!  Starting from the beginnings of
geometry as a deductive science in the {\it Elements} of Euclid, continuing in the
attempts to free Euclid of perceived flaws (especially regarding the status
of Postulate V), through Bolyai and Lobachevsky and their flights of imagination,
then the development of geometry through the introduction of 
coordinates and tools from calculus, we have seen how the history of the subject
has been shaped by what went before.  I hope you have a bit more of an
appreciation of how the mathematics of the present connects with the 
questions that began the study of the subject, and how mathematics
is really a very unified subject, where the neat divisions into algebra, 
analysis, and geometry are actually pretty arbitrary.  When it comes to 
understanding the features of the mathematical world, any tool is potentially
useful and it's really ``all of a piece.''


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