THIS IS PART II OF DR. STRUPPA'S PRESENTATION ON RIEMANN NON-EUCLIDEAN 
GEOMETRY.


4. Riemann's life.

There are many books which contain detailed accounts of Riemann's life 
and mathematical contributions. My main purpose here is to give you a 
rough idea of what his place is among mathematicians, and I would be very 
happy if you would find the time to browse a book such as Bell's Men of 
Mathematics (or any other book in the history of mathematics) to get a 
richer perspective on Riemann.

Bernard Riemann was born near Hannover (Germany) from a poor family in 
1826; he obtained his Ph.D. in Gottingen. His dissertation was a landmark 
work in the theory of one complex variable, and was highly praised by 
Gauss (which we have already mentioned throughout the course as one of 
the greatest mathematicians ever). In 1854, Riemann was 
nominated Privatdozent at that same university. At those times, being a 
Privatdozent did not guarantee a salary, since Professors were paid 
depending on the enrollment of their classes (something to think about, 
you may say!). From that period we have a very moving letter of Riemann 
to his father, in which he expresses happiness because his class actually 
consisted of 8 students and he was not expecting such a large number (and 
the relative higher than expected income).

An important part of becoming Privatdozent was the necessity for the new 
professor, to give a presentation which could be accessible to the entire 
faculty of the university. This is another lost tradition which, it seems 
to me, should be resurrected. In this particular case, Riemann provided 
us with one of the great masterpieces of all mathematics, an essay titled 
"On the hypotheses which lie at the foundations of geometry". We will 
come back later on to the content of this paper, wihch deals with the 
possibility of founding geometry on the notion of "measurement", and of 
distance.

Riemann was very well considered by the mathematicians of his age (in 
particular Gauss, who was in the last years of his life, and Dirichlet, 
who had succeeded Gauss at his Chair in Mathematics), and in 1857 he 
became Assistant Professor and finally Full Processor in 1859, on the 
Chair which had been earlier of Gauss and Dirichlet.

His health was unfortunately quite bad, as he suffered from TBC and he 
died in northern Italy in 1866, in a small town on the Lago Maggiore.


5. Riemann's non-Euclidean geometry.

We are finally ready to attempt a brief description of the way in which 
Riemann dealt with the OAH, to construct his non-Euclidean geometry.

I will not be able to reproduce the historical sequence of ideas and 
resutls which lead to Riemann's construction. I will, however, hopefully 
provide a sufficiently clear intellectual sequence, which may not be 
historically correct, but which explains and motivates the final results. 
Also, I wish to remind you that this particular part of my work would 
require (to be dealt with precision) the use of substantial tools from 
differential calculus. We do not have them, and I will therefore rely 
heavily on your intuition.

It is probably fair to say that the starting point for the creation of 
Riemann's non-Euclidean geometry was his interest for the notion of 
"distance"; we all know that if we consider a point (x,y) in the 
cartesian coordinate plane, then its distance from the origin is given by 
the Pythagorean theorem as

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On the other hand, there may be situations in which this notion of 
distance may not be appropriate to the problems at hand. Consider, for 
example, the case of a taxi driver in New York City. We can model 
downtown Manhattan with a grid of perpendicular lines, in which the 
vertical ones are the Avenues, and the horizontal ones are the Streets.
If we now fix the origin at the intersection of Fifth Avenue (out of 
respect for the Fifth Postulate) with 36th Street, and if we assume that 
each block (regardless of its orientation) is of length 1 (in some unit), 
then we see that the distance a taxi driver has to drive to reach the 
intersection of Third Avenue with 42nd Street is not given (as one would
get from Pythagora's theorem) by the square root of 40, but rather by 8 
(where 8 is indeed the number of blocks a driver must drive to get from 
one point to the other).

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It is clear therefore that in this way we have defined a different way of 
measuring distances, which in some situations may be closer to the 
practical needs (and/or may better describe the geometry of the space). 
In this case, the example may seem artificial because we "see" the 
buildings as obstacles to the shortest way, but what we are doing is in 
fact rather deep, because one may think of the "distance" as the way to 
express intrinsic properties of the space. The most significant example 
that comes to mind is the so-called space-time of Relativity Theory. In 
this case (you may want to look at any introductory exposition of 
relativity theory), the points have really four coordinates: three which 
indicate their spatial position, and one which indicates their temporal 
position. Thus, if we want to describe an even which takes place in a 
point of coordinates (1,3,2) at time t=5, we will say that the point has 
space-time coordinates (1,3,2,5). In relativity theory, such a point is 
called an "event" (this makes sense because a point is now a location 
and a time, like saying, Room 203B in S&TI at 3.00 pm; this clearly 
corresponds to what we would call an event in everyday language). Now, the 
question of understanding what is the "distance" between events is not 
simple anymore. While we can assume to know the distance between two 
points (again Pythagora!) or between two instant of times (the "distance" 
between 10.30 and 11.15 is 45 minutes), it becomes quite a different 
thing to understand what is the distance between two events. In fact, 
deciding what this distance would be amounts to describing the "shape" 
or the "geometry" of the space-time we live in (recall here our lengthy 
discussions on Flatland and light-rays).

It is one of the fundamental point of relativity theory the fact that 
such a distance can be described by a weird variation of Pythagora's 
formulas. In fact, the relativistic distance between an event (x,y,z,t) 
and the origin (0,0,0,0), can be given by the square root of


                         2  2  2  2
			x +y +z -t 


The precise meaning of this definition is too complex for us, but it is 
sort of important to see that several different definitions can indeed be 
given.

All of this is very good as long as we remain on the plane or on a space 
such as in the case of relativity theorem; but what if we want to play a 
similar game on a surface?

One of the greatest contributions of Riemann (in his 1854 paper) was to 
show how it is possible to define a similar notion of distance on 
(almost) arbitrary surfaces. In so doing, Riemann noticed the relevance, 
for his theory, of the notion of geodesics. Let us spend a few lines on 
this concept. When we look at two points on a plane, the segment of line 
which connects them has the quality of being the shortest path between 
them, at least if we use the usual notion of distance; similarly, for a 
cab driver, there are similar notions of shortest paths, which do not 
necessarily give rise to straight lines. One can also see that in 
relativity theory the shortest parth is not a straight line in the usual 
sense. So, what Riemann realized, is that the reason why lines play such 
a crucial role in Euclidean geometry is because of this "metric" 
property, and he therefore proposed to look at various geometries on 
surfaces, where the word "line" had now to be interpreted as "geodesic", 
i.e. arc of shortest length between two points.

In the simplest examples, a geodesic in a plane is just a segment, while 
a geodesic on the sphere is an arc of greatest circle. You may want to 
try to figure out what is a geodesic on a cylinder, or on a torus (this 
is the technical name for a donut).  

Riemann therefore proposed a new type of geometry in which the plane is 
replaced by any surface, points are still points, but straight lines are 
replaced by geodesics and, similarly, segments are replaced by arcs of 
geodesics. One of the first, natural, questions is the following: what 
can we say about the geometries which we can construct on different 
surfaces? Are they somewhat equivalent? In particular, are they all 
euclidean geometries (in the sense that they satisfy the fifth euclidean 
postulate)? It turns out, interestingly enough, that the answer to the 
last two questions is negative. Geometries constructed on different 
surfaces need not be equivalent, and actually may not be euclidean.

To qualify this statement, one needs to take one step back, and look at a 
fundamental idea of Gauss (which we will need later on when we will 
discuss about map making and cartography). 

What gauss did was to introduce the notion of "curvature" of a surface. 
This is too much of a technical notion for us to describe in detail, so I 
will limit myself to a rough idea. Consider a surface and a point P on 
it. We can now cut the surface along P in such a way as to obtain the 
"most curved" section we can (intuitively). We can then cut the same 
surface along the direction perpendicular to the first cut. In this way


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we obtain two curves (when you cut a surface with a plane, the profile 
you get is a curve) which both contain P. There are now thre 
possibilities: one of the two curves is a line (for example if the 
surface is a cylinder);

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the point P is a maximum point or a minimum point for both curves (for 
example if the surface is a sphere);

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or the point P is a maximum point for a curve and a minimum point for the 
other curve (for example if the surface is what we call a pseudosphere).

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In these three cases we will respectively say that the curvature of the 
surface, AT THE POINT P, is zero, positive or negative. Please pay 
attention to the fact that we are not giving a value to the curvature 
(doing so is possible but much more complicated) but we are only deciding 
whether the curvature is positive, negative or zero. The other important 
point to notice is the fact that the curvature may change from point to 
point. In other words (and you are invited to think of examples) a 
surface may have points of positive curvature, points of zero curvature 
and points of negative curvature. The three surfaces we have used as 
examples (cylinder, sphere and pseudosphere) are three examples of 
"constant curvature" surfaces, i.e. surfaces in which the curvature is 
the same for every point; even though they are our primary examples, you 
should keep in mind that they are somehow very special cases. As a matter 
of notation we will say that a surface of zero curvature is a "flat" 
surface.

One of the fundamental results of Gauss (to which we will return later 
on) is the fact that one cannot apply a surface to another one of 
different curvature (thus it is, for example, impossible to make a 
perfect map of the world, since a map is a plane object, and therefore it 
has zero curvature, while the world is a sphere, and therefore it has 
positive curvature).

Before we state our next fundamental result (due to gauss as well), we 
would need a short excursion into radians and degrees. Since you have 
seen these notions in class with Dr. Gabel, you may want to supply the 
missing part by taking it from your own notes.

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We are now ready for the following statement:

Theorem. If we draw a triangle on a surface of constant curvature K, then 
the product of the area of the triangle times K equals the difference 
between the sum of the angles of the triangles and pi (the radian way of 
measuring two right angles). In symbols, if we consider a triangle ABC 
whose angles we denote by a,b,c (the measures being in radians), we have

			Area(ABC)*K=a+b+c-pi.

There are some immediate observations that can be made from this theorem; 
first we note that the sum of the angles of a triangle is 180 degrees if 
and only if K is zero. Thus euclidean geometry is the geodesic geometry 
on a flat surface!

Obviously, this theorem also allows us to go back to our AAH and OAH and 
discuss what kind of geometries they provide us.

Indeed, since the area of a triangle is always a positive number, we see 
that AAH is equivalent to requiring that K<0. Thus, the pseudosphere will 
be the model for AAH-geometries (as the Gliders will show). Analogously, 
OAH is clearly equivalent to a geodesic geometry on a surface with 
positive curvature.

But how can we reconcile this "new result" with what we had obtained 
earlier on? If you recall, we saw that Saccheri conclusively (and 
correctly) proved that if we accept the first four axioms in Euclid 
construction, then the OAH was not acceptable. Does this contradict our 
new result?

In order to answer this question, let us take a closer look at the new 
non-Euclidean geometry which we have constructed. This is the geometry of 
geodesics on a sphere. In this geometry, points are still points, but 
lines must be interpreted as great circles (these are the geodesics) and 
segments as arcs of great circles. We therefore see that the axiom that 
only one line should pass through two fixed points is now violated (just 
think of the north pole and of the south pole; these are two points, and 
yet every meridian -clearly a great circle- passes through each of them); 
in addition, also the axiom on the possibility of indefinitely continuing 
a straigth line is now being violated; lines do not have anymore infinite 
length in this geometry. So we are now reconciling this vision with 
Saccheri's. He was indeed right in claiming that OAH was unacceptable 
within the confines of the first four Euclidean axioms; it is only if we 
accept to violate two more axioms (in particular, the one on the 
continuation of the straight line, since we could take some steps to 
still verify the first one), that we see the possibility of dealing with 
an OAH geometry. 

6. Conclusions and the matrix.

Now that we have finished a rather arduous journey, let su try to look 
back in retrospective and see how all of this fits within the matrix 
which is the backbone of the course. I iwll only provide a few hints, 
here, of how these discussions fit within our matrix. I think it would be 
of great importance that you try to discover for yourselves other aspects 
which I have overlooked.

6.1. Tensions.

We certainly have a finite-infinite tension, which 
is particularly clear in the notion of parallel itself; in order to even 
define the notion of parallel, one necessitates the use of infinity (such 
as in ...lines which neve meet...); still the pictures we are able to 
draw are by necessity finite, and this conflict between finite 
representation and infinite "expectation" is at the core of the problem. 
Analogously, the conflict between finite and infinite is evident in the 
axiom which requires a line to be indefinitely extendable. The advent of 
Riemann's non-Euclidean geometry sheds new light on this tension.

The pure-applied tension is also a part of this topic, as it relates (see 
our lengthy discussion in class) to the nature itself of an axiomatic system.

6.2. Cultural Context

The world-view is at stake in this particular anchor and topic. What is 
the geometry of the world? Do we live in a Euclidean world? We have no 
answer to this puzzling question, and much of the appeal of the subject 
seem to be in this uncertainty.

6.3. Mathematical Perspective

Measurement seems to me to be the key perspective here. How do we know 
which world are we in? The only way is to measure angles, and to make 
sure we have sufficient precision when we do this. So there is an 
interesting link between perspectives and cultural contexts. This also 
relates to the notion of approximation, as the precision of the 
measurement is here a key to the decidability of the geometry we live in. 
Please try to take the time to think some more about this.