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What is Hyperbolic Geometry?*

Mahan Mj

Abstract. We give a brief introduction to hyperbolicgeometry, including its genesis. Some familiarity withhigh school calculus and co-ordinate geometry is allthat is assumed and most of the article should, inprinciple, be accessible to beginning undergraduates.

1. Introduction

1.1. History

Euclid was the first to formalise geometry into anaxiomatic system. One of his axioms called the

parallel postulate has been the focus of a lot ofmathematical attention and work for almost twomillennia. It states:Given a straight line L and a point x outsideit, there exists a unique straight line L passingthrough x and parallel to L.

This postulate has certainly existed from atleast as far back as 200 BC and much effortwent in to try to prove it from the other ax-ioms of Euclid. In ancient times, Proclus, OmarKhayyam, Witelo, Gersonides, Alfonso, amongstothers made the attempt. In more recent times,

Saccheri, Wallis, Lambert, and even Legendrefailed in this attempt, with good reason as wehope to show in this article.

To appreciate what this problem means, wefirst state Euclids axioms (appearing in the firstbook of Elements).

(1) A straight line may be drawn from any pointto any other point.

(2) A finite straight line may be extended con-tinuously in a straight line.

(3) A circle may be drawn with any centre andany radius.

(4) All right angles are equal.(5) If a straight line falling on two straight lines

makes the interior angles on the same sideless than two right angles, the two straightlines, if extended indefinitely, meet on theside on which the angles are less than tworight angles.

It is the fifth postulate that is equivalent tothe parallel postulate stated above. The reasonwhy people tried to prove it from the rest of theaxioms is that they thought it was not sufficiently

This is an expanded and revised version of an article with

the same title that appeared in the Ramanujan MathematicalSociety Newsletter Vol. 19 No. 2, September 2009. We aregrateful to the Ramanujan Mathematical Society for allowingus to submit the revised version to APMN.

self-evident to be given the status of an axiom,and an axiom in Euclids times was a self-evident truth. Almost two millennia passed withseveral people trying to prove the fifth postulateand failing.

Gauss started thinking of parallels about 1792.In an 1824 letter to F A Taurinus, he wrote:The assumption that the sum of the three an-gles (of a triangle) is smaller than two right an-gles leads to a geometry which is quite differentfrom our (Euclidean) geometry, but which is initself completely consistent. But Gauss did notpublish his work. Already in the 18th century,Johann Heinrich Lambert introduced what are to-day called hyperbolic functions and computed thearea of a hyperbolic triangle. In the 19th century,hyperbolic geometry was extensively exploredby the Hungarian mathematician Janos Bolyaiand the Russian mathematician Nikolai IvanovichLobachevsky, after whom it is sometimes named.Lobachevsky published a paper entitled On theprinciples of geometry in 18291830, while Bolyaidiscovered hyperbolic geometry and publishedhis independent account of non-Euclidean geome-

try in the paper The absolute science of space in 1832.The term hyperbolic geometry was introducedby Felix Klein in 1871. See [2] and [4] for furtherdetails on history.

To get back to our article, we restate theparallel postulate, expanding it somewhat andunderscore the terms we will investigate.

Postulate 1.1. (Parallel Postulate) Given a straightline L in a plane P and a point x on the plane Plying outside the line L, there exists a unique straightline L lying on P passing through x and parallelto L.

Then the problem we address is:

Problem 1.2. Prove the Parallel Postulate 1.1 fromthe other axioms of Euclidean geometry.

To properly appreciate the (rather unexpected)solution to Problem 1.2 we need to investigate thefollowing keywords more thoroughly:

(1) straight line(2) plane(3) parallel.

1.2. Terms and definitions

Definition 1.3. The Euclidean plane is R2

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Asia Pacific Mathematics Newsletter


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equipped with the metric

ds2 = dx2 + dy2.

We have suddenly sprung the new concept ofa metric on the unsuspecting reader. A word ofclarification is necessary to justify our assurancein the Abstract that some familiarity with highschool calculus and co-ordinate geometry is allthat is assumed. We all know that distances inthe Cartesian plane are measured by the formula

d((x1,y1), (x2,y2)) = [(x1 x2)2+ (y1 y2)

2]12

for points (x1,y1), (x2,y2) R2.

Definition 1.3 is just the infinitesimal formof this formula. The reason why it is called ametric is that it provides us a means of measuringlengths of curves (thought of as smooth maps of[0, 1] into R2) according to the formula

l() =1

0ds =

10

dxdt2+

dydt

212

dt (A)

for some parametrisation x = x(t),y = y(t) of thecurve .

We now turn to the notion of a straight line inthis context. A fact we are quite familiar with, butwhose proof is not quite trivial, is the following.

Proposition 1.4. Given two points (x1,y1), (x2,y2) R

2, the straight line segment between (x1,y1) and(x2,y2) is the unique path that realises the shortestdistance (as per formula A) between them.

We shall not prove this here, but when weturn to hyperbolic geometry we shall give theproof of a corresponding statement in hyperbolicgeometry and leave the reader to modify it ap-propriately to prove Proposition 1.4. Note thatthe term straight line segment in Proposition 1.4means straight line in the sense of Euclid.

Finally we give the precise meaning of theword parallel.

Definition 1.5. Two bi-infinite straight lines aresaid to be parallel if they do not intersect.

Here, by a bi-infinite straight line we meanthe result of extending a straight line segment in-finitely in either direction, so that any subsegmentof it is a straight line.

Note: All the notions above boil down to thesingle notion of a metric in Definition 1.3. It isstriking that all the features of Euclidean geom-etry have this single structural definition at theirfoundation. In the language of logic, R2 equippedwith the metric

ds2 = dx2 + dy2

is called a model for Euclidean geometry. We shall

not go into this, though the real solution to Prob-lem 1.2 involves at least an implicit investigationof these foundational issues.

2. Metric Geometry

In this article, we shall be interested in a some-what more general form of a metric than Defini-tion 1.3. This will give us quite a general classof models. We state this somewhat informallybelow.

Definition 2.1. A metric on an open subset U ofthe Euclidean plane R2 is a method of computingarc lengths l() as per the formula

ds2 = f(x,y)dx2 +g(x,y)dy2.

Thus the length of will be given by the formula

l() =

10

ds =

10

f(x(t),y(t))

dx

dt

2

+g(x(t),y(t))

dy

dt

2

12

dt (B)

for some parametrisation x = x(t),y = y(t) of thecurve .

An isometry I is a map that preserves themetric, i.e. if I((x,y)) = (x1,y1) then

f(x,y)dx2 +g(x,y)dy2 = f(x1,y1)dx21 +g(x1,y1)dy

21.

The property of straight lines given by Propo-sition 1.4 is then abstracted out to give the nextdefinition.

Definition 2.2. An arc in U equipped with themetric above is said to be a geodesic if it locallyminimises distances, i.e. there exists > 0 suchthat for a, b ([0, 1]), if there is some arc oflength at most connecting a, b, then the subarcof joining a, b is the unique path that realisesthe shortest distance (as per formula (B)) betweenthem.

The definition of parallel lines 2.3 goes throughin this more general context by just replacing theterm straight line by the term geodesic.

Definition 2.3. Two bi-infinite geodesics are saidto be parallel if they do not intersect.

3. Hyperbolic Geometry

3.1. The model

A Model for hyperbolic geometry is the upperhalf plane H = (x,y) R2,y > 0 equipped with themetric

ds2 =1

y2(dx2 + dy2). (C)

H is called the Poincare upper half plane inhonour of the great French mathematician whodiscovered it.

Motivation, an aside: Without any motivation,the model described above seems to have comeout of thin air. This is the only place where, during

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the course of this article, some more mathematicalbackground is necessary to fully appreciate whatis going on. The young reader is therefore encour-aged to take the above model at face value andskip this motivation.

Poincare who proposed the above model, cameto it from complex analysis. It is a fact that any

simply connected complex 1-manifold (whateverthat means) is essentially one of the followingthree:

(1) The Riemann sphere C(2) The complex plane C(3) The upper half plane H = {z C : Im(z) > 0}.

Poincare came to hyperbolic geometry fromcomplex analysis and, suffice to say, in this contextthe model for H is quite natural as one of threepossible models for a simply connected complex1-manifold.

3.2. Geodesics and isometries

It is not hard to see that vertical straight lines(given by x = k, a constant) are geodesics (seeDefinition 2.2). Consider two points a = (k, u) andb = (k, v) in H. Then any path from a to bthat is not vertical will have nonzero dx

dtat some

point. Then its length computed as per formula(C) will be strictly greater than that of the pathP(), where P is the projection onto the verticalline x = k, given by P(x,y) = (k,y). Hence as perDefinition 2.2, we have

Lemma 3.1. Vertical straight lines in H are geodesics.In fact, the vertical segment between a, b is the uniquegeodesic between a, b.

Now, we compute some explicit isometries(see Definition 2.2).

Lemma 3.2. Define f : H H by f(x,y) = (x + a,y)for some fixed a R.Define g : H H by g(z) = R

2

zfor some fixed R > 0,

where z denotes the complex conjugate of z.Then f,g are isometries ofH.

Proof. First note that f is given in real co-ordinatesand g is given in complex co-ordinates. To checkthat f is an isometry, put x1 = x + a, y1 = y. Then(since a is a constant) 1

y2(dx2 + dy2) = 1

y21

(dx21+ dy2

1)

and hence f preserves the metric given by formula(C), i.e. f is an isometry.

Next, to show that g is an isometry, we expressformula (C) in complex co-ordinates as

ds2 =dzdz

Im(z)2.

It is a simple calculation to show that if we put

z1 = g(z) =R2

z, then dzdz

Im(z)2=

dz1dz1Im(z1)2

. This shows that

g is an isometry. Maps such as f are called parabolic translations

and those like g are called inversions. The map

g is an inversion about a semi-circle of radius Rcentred at 0. A more geometric description of gis as follows. Take a circle of radius R centredat 0. Then every point in H lies on a unique raythrough 0. Let p H be on some such ray r. Theng(p) is the unique point on r for which the productof the radial co-ordinates of p and g(p) is R2.

We shall have need only for inversions in whatfollows. These maps are the hyperbolic geome-try analogues of reflections in plane Euclideangeometry.

There is nothing special about the point 0 inLemma 3.2 for the map g. We might as well shiftthe origin 0 to a point p on the real axis and con-sider inversions about semi-circles centred at p.Then essentially the same computation shows thatinversions about semi-circles of arbitrary radius Rcentred on the real line are isometries. The readermay refer to [1] and [3] for similar computations.

Next, from Definition 2.2, it follows that theimage of a geodesic under an isometry is anothergeodesic. Hence images of vertical geodesics un-der inversions are geodesics.

Now consider a semi-circle C1 with centreat (a, 0) with radius R. Consider the semi-circleC2 with centre at (a + R, 0) with radius 2R. Letinversion in C2 be denoted by I2. Let l denotethe vertical geodesic x = (a R). Then it is asimple exercise in Euclidean geometry to showthat I2(l) = C1. We have thus finally establishedthe following.

Theorem 3.3. In the hyperbolic plane H, verticalstraight lines and semi-circular arcs with centre onthe real axis are geodesics.

Let L denote the collection of vertical straightlines and semi-circular arcs with centre on the realaxis. Given any l L and any point p outside l onH there are infinitely many l L through p suchthat l l = .

We are finally in a position to answer Question1.2 adequately. There are models (H in the abovedescription) which satisfy all axioms of Euclideangeometry except the parallel postulate.

Theorem 3.4. We have the following New ParallelPostulate:Given a geodesic L in H and a point x on H lying out-side L, there exist infinitely many bi-infinite geodesicsL lying on H passing through x and parallel to L.

The careful reader will notice that we haveonly violated the uniqueness part of the parallelpostulate in constructing H. Existence continuesto hold.

To end this section we mention the fact thatthe collection L described above is the collec-

tion of all geodesics in H. The interested readercan prove this using the uniqueness part ofLemma 3.1.




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4. Fuchsian Groups and ClosedHyperbolic Surfaces

We now come to the study of discrete subgroupsof the isometry group of the hyperbolic plane.Poincare initiated the study of these groups. Anoctagon with suitable side identifications (see

Diagram 1 below) gives a genus two surfacetopologically.

Diagram1

This is just a topological picture. We want toconvert the flabby topological information into amore precise geometric picture. We first suppose

that the above octagon is hyperbolic, i.e. it isisometric to a piece of the hyperbolic plane. Toglue it as per the recipe given in the Diagram 1above, we need to ensure two things:

(1) The sides labelled by the same letters aregeodesics of equal length.

(2) The angle sum obtained at the vertex towhich all the eight vertices of the octagonare identified is equal to 2.

Towards this, we first consider a more sym-metric model of the hyperbolic plane, the Poincaredisk. This is the unit disk in the complex plane

with the metric 4|dz|2

(1|z|2

)2 . It turns out that the

Poincare disk is isometric to the upper half planewith the hyperbolic metric.

Now consider an ideal hyperbolic octagon, i.e.an octagon, all whose vertices are on the circleat infinity. See Diagram 2.

The geodesics in this model are semi-circlesmeeting the boundary circle at right angles. Sinceadjacent sides (geodesics) both meet the boundarycircle at right angles, it follows that the internalangles of an ideal octagon are all zero. Nowstart shrinking the octagon slowly towards a verysmall regular octagon in the Poincare disk, ensur-

ing that the octagon is symmetric about the origin.Then a very small regular hyperbolic octagon hasinternal angles very close to the internal angles

Diagram2

of a regular Euclidean octagon, which is

3

4 . Itfollows from the intermediate value theorem thatat some intermediate stage, all internal anglesequal 4 . Take the regular hyperbolic octagon withall internal angles equal to

4and glue alternate

sides as per the recipe in Diagram 1.Since all sides are equal, it follows that sides

labelled by the same letters are geodesics of equallength. The angle sum obtained at the vertex towhich all the eight vertices of the octagon areidentified is equal to 8

4 =2. Thus the identi-

fication space is a hyperbolic genus two surface.The isometries of the hyperbolic plane that iden-

tify sides labelled by the same letter (preservingorientation) generate a discrete subgroup of theisometry group of the hyperbolic plane, such thatthe quotient of the hyperbolic plane by this groupis the hyperbolic genus two surface.

The above construction is immensely flexible.We can vary lengths of sides and angles ensuringonly the conditions that sides labelled by the sameletters are geodesics of equal length and that theangle sum obtained at the vertex to which allthe eight vertices of the octagon are identifiedis equal to 2. Thus we get continuously many

hyperbolic genus two surfaces by varying theabove construction.

5. Kleinian Groups and Thurstons Work

In one dimension higher, we have hyperbolic3-space H3 = {(x,y,z) : z > 0} equipped with the

metric ds2 =dx2+dy2+dz2

z2. Discrete subgroups of the

isometry group of hyperbolic 3-space are termedKleinian Groups. As in the two dimensional case,one can construct an all-right hyperbolic dodec-ahedron, i.e. a dodecahedron in hyperbolic space

such that(1) All faces are totally geodesic regular hyper-

bolic pentagons.

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(2) All dihedral angles are right angles.

Gluing opposite faces of the dodecahedron bya three-fifths twist we obtain a closed hyperbolic3-manifold called the SeifertWeber dodecahedralspace. The group of face-pairing transformationsgenerate a Kleinian group such that the quotientof hyperbolic 3-space by this group is the Seifert

Weber dodecahedral space.Thurstons Geometrisation Conjecture (proved

by Perelman) states roughly that a generic topo-logical 3-manifold is hyperbolic. A slightly moreprecise formulation is that any topological 3-manifold can be canonically decomposed alongspheres and tori such that each resulting piece car-ries a geometric structure. Also amongst geomet-ric structures, a hyperbolic structure is generic.The celebrated Poincare Conjecture is a smallcomponent of this conjecture. The first impetustowards this conjecture was given by Thurston

himself, who, in his Fields Medal winning workshowed that a vast collection of topological

3-manifolds (the so-called Haken atoroidal ones)carry a hyperbolic structure. We refer the readerto [5] for this, amongst several other conjectures ofThurston. Almost all these conjectures have nowbeen resolved affirmatively after three decades ofhard work by a number of mathematicians atribute to the great insight that Thurston had.

References

[1] R. Benedetti and C. Petronio, Lectures on HyperbolicGeometry, Springer Universitext (1992).

[2] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds,2nd Edition (GTM Springer, 2005).

[3] G. P. Scott, The geometries of 3-manifolds, Bull.London Math. Soc. 15 (1983) 401487.

[4] J. Stillwell, Sources in Hyperbolic Geometry, Volume10 in AMS/LMS series History of Mathematics(1996).

[5] W. P. Thurston, Three dimensional manifolds,Kleinian groups and hyperbolic geometry, Bull.

Amer. Math. Soc. (1982) 357382.

Mahan MjRKM Vivekananda University, India

Mahan Mj is a Professor of Mathematics at School of Mathematical Sciences,

RKM Vivekananda University.



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