ńņš. 8 |

the book were homogeneous under appropriate groups. For example, the Euclidean

geneous

group acts transitively on En : any point of En goes to the origin under a suitable

spaces

Euclidean motion. The afļ¬ne group Aff(n) acts transitively on pairs of distinct points

of An ; as discussed at several points of the book, this is closely related to the fact that

afļ¬ne geometry does not have an invariant distance function.

If is a G-set and x ā , the stabiliser subgroup of x is the set of elements of G

that ļ¬x x, that is

StabG (x) = h ā G h(x) = x .

For example, the stabiliser subgroup of the origin 0 ā En in Eucl(n) is the group O(n)

of orthogonal matrixes.

If G acts transitively on , the map ex : G ā’ deļ¬ned by g ā’ gx is surjective.

Moreover elements g1 , g2 ā G map to the same point of if and only if g2 = g1 h

for some h ā StabG (x); thus ex induces a bijection G/ StabG (x) ā’ . (Here G/H

ā’

stands for the quotient of G by the equivalence relation g ā¼ gh for h ā H , or the set

of left cosets of H .)

A homogeneous space under G is a principal homogeneous space

Definition

under G or a G-torsor if the stabiliser StabG (x) is trivial for every x ā . Since the

stabilisers of x and gx are conjugate (by the same argument as in Exercise 6.7), it is

enough to verify that StabG (x) is trivial for a single x ā .

For example, afļ¬ne space An is a homogeneous space under Aff(n), but is a torsor

under the translation subgroup Rn ā‚ Aff(n).

According to the previous discussion, if is a G-torsor, then ex : G ā’ is a

bijection from G to , and I could use this to identify G and . However, differ-

ent elements of give different bijections: the set has no distinguished identity

element.

Let consist of the vertexes of a regular n-gon in the plane E2 , G ā‚

Example

Eucl(2) the group of symmetries of (the dihedral group D2n , see Exercise 6.5),

and let H be the cyclic subgroup of G of order n consisting of rotations. (Draw a

picture!) Then the geometric action of G on is transitive, since the polygon is

regular. Thus is a homogeneous space under G. The stabiliser StabG (P) of a vertex

P ā is of order two, consisting of the identity and the reļ¬‚ection in the axis through

P. The subgroup H acts transitively and without stabilisers (since it does not contain

reļ¬‚ections). Thus is an H -torsor: there are as many vertexes as rotations, but no

vertex is distinguished over the others.

9.2 GROUP THEORY 171

9.2.3 Recall Kleinā™s Erlangen program of Section 6.3: the slogan is that geometry is the study

The of properties invariant under a transformation group G. The introduction to Chapter 1

Erlangen discussed the basic geometric and philosophical principles: space should be

program

revisited (1) homogeneous (the same viewed from every point), and

(2) isotropic (the same in every direction).

In terms of the group of transformations, (1) says that the group G acts transitively

on points of space, whereas (2) says that it also acts transitively on coordinate frames

based at every point. Helmholtzā™ axiom of free mobility requires slightly more: it

also says that, given two points of the space and sets of coordinate frames based at

these points, there is a unique element of G mapping one to another. In other words,

the set of all coordinate frames at all points is a G-torsor (principal homogeneous

space under G). Thus

r Euclidean space En is a homogeneous space under the Euclidean group Eucl(n). The

stabiliser of a point P ā En is isomorphic to the group O(n), the group of rotations

and reļ¬‚ections ļ¬xing P. By Theorem 1.12, the set of Euclidean frames forms a torsor

under Eucl(n).

r The sphere S n is a homogeneous space under the group O(n + 1) of spherical motions

(Theorem 3.4 for n = 2; the general case is identical). For P ā S n , the stabiliser group

is isomorphic to the group O(n). (It is the group of orthogonal matrixes in the Rn that

is the orthogonal complement of O P.)

r Hyperbolic space Hn is homogeneous under the Lorentz group O+ (n, 1). The sta-

biliser of a point P is again isomorphic to the group O(n).

r Projective space Pn is homogeneous under the projective linear group PGL(n + 1).

The stabiliser of a point P ā Pn is PGL(n). By Theorem 5.5, the set of projective

frames of reference forms a PGL(n + 1)-torsor.

The notion of torsor formalises the ad hoc deļ¬nition of afļ¬ne space I gave in Chapter 4.

9.2.4

Let V be a vector space; an afļ¬ne space A(V ) is just a torsor under V . In other words,

Affine space

A(V ) is a set with an action of V (ā˜by translationā™), and this action is simply transitive:

as a torsor

for P, Q ā V there is a unique vector x ā V such that Q = P + x.

Looking back to 6.5.3, I can say all this slightly differently: the transformation

groups in Euclidean and afļ¬ne geometry are semidirect products. For example, the

Euclidean group

Eucl(n) = O(n) Rn

is the semidirect product of the normal subgroup of translations and the group of

rotations. From the analysis of 6.5.3, it follows that the subgroup O(n) is not normal.

The conjugation construction (see 6.4) allows me to deļ¬ne Euclidean space to be the

space of all conjugates of a ļ¬xed copy of O(n) ā‚ Eucl(n), and notions of Euclidean

geometry to be all notions that can be deļ¬ned on this space invariantly under the

group Eucl(n). This is of course the Erlangen program repeated once again.

172 CONCLUDING REMARKS

I can say the same words starting from the group of afļ¬ne transformations Aff(V )

(see 4.5). This contains copies of GL(V ), the group of invertible linear maps of V , as

afļ¬ne transformations ļ¬xing a point, and these subgroups are once again nonnormal.

From the group theory it follows then that the group of translations V acts transi-

tively with trivial stabiliser on A(V ); thus A(V ) is a V -torsor (a principal homogeneous

space under the group of translations). In other words, we have an action Ļ•v : P ā’

P + v of the additive group of V deļ¬ned on points of afļ¬ne space. For P ā A(V ),

we get a bijection e P : V ā’ A(V ) mapping v ā V to P + v; two such identiļ¬ca-

tions differ by an element of V acting by translation. The bijections e P are differ-

ent coordinate systems on afļ¬ne space, differing by a translation; in the coordinate

system e P , the point P plays the role of origin. We also see that two points

ā’ā’

P, Q ā A(V ) determine a vector e P (Q) = P Q ā V (cf. Figure 4.2).

The point here is that for the cases I am interested in, I can recover the geometry

from the group or the group from the geometry. For example, if the Euclidean group

Eucl(n) and its subgroup O(n) are given, En is the homogeneous space Eucl(n)/ O(n),

where O(n) = Stab(x); alternatively, En is the set of subgroups conjugate to O(n).

9.3 Geometry in physics

Some of the most substantial applications of geometric ideas come from physics.

Recall the grandiose aim expressed in my ļ¬rst sentence:

Geometry attempts to describe and understand space around us and all that is in it.

You may well object that most of the work so far has gone into describing the space, so

it is about time I told you something about what is in it. The discussion is necessarily

somewhat sketchy and in places wildly over-simpliļ¬ed; at the end I give references

to the literature for further study.

The dynamics of Galileo and Newton takes Euclidean three space E3 as the funda-

9.3.1

mental model of physical space, and time t as a universal parameter with a preferred

The Galilean

directionality. Thus spacetime is modelled by E3 Ć— R, with coordinates (x, t). Spatial

group and

lengths are measured with respect to the Euclidean metric of 1.1, and involve only the

Newtonian

x-coordinate; events also have a time separation t2 ā’ t1 (no absolute value is taken

dynamics

here). Valid coordinate systems describing Newtonian dynamics are based on inertial

frames in uniform relative motion with respect to each other, in which spatial lengths

and time differences are unchanged. Transformations to a different coordinate system

are therefore given by maps

(x, t) ā’ (Ax + gt + b, t + s),

where A ā O(3) is a 3 Ć— 3 orthogonal matrix, g and b are 3 Ć— 1 column vectors, and

s ā R is a scalar. Such transformations collectively form the Galilean group Gal(3, 1)

of classical (3 + 1)-dimensional spacetime E3 Ć— R. A simple parameter count shows

that the Galilean group depends on 3 + 3 + 3 + 1 = 10 parameters. You recognise

Eucl(3) as a subgroup of Gal(3, 1) consisting of time-independent transformations

9.3 GEOMETRY IN PHYSICS 173

Table 9.3 Symmetries and conservation laws

Conserved

Symmetry quantity Name

dxi

spatial translation mi momentum

dt

(x, t) ā’ (x + b, t) i

dxi

m i xi Ć—

spatial rotation angular momentum

dt

(x, t) ā’ (Ax, t) i

ā’pt +

Galilean boost m i xi centre of mass (where

(x, t) ā’ (x + gt, t) i p is the total momen-

tum)

1 dxi 2

time translation mi energy

2 dt

(x, t) ā’ (x, t + s) i

(x, t) ā’ (Ax + b, t),

with g = 0 and s = 0. Transformations with nonzero g correspond to a change to a

new reference frame in uniform movement of speed g with respect to the old one; such

group elements are usually called Galilean boosts. Elements of Gal(3, 1) with s = 0

correspond to moving the origin of time; Newtonian physics has no ļ¬xed Creation

or Big Bang. It is however not possible to stretch or reverse time, however much you

might wish it during an exam.

The shape of the Galilean group determines Newtonā™s equation of motion, in the

form familiar to you from a ļ¬rst mechanics course. For a single particle with mass m

and position vector x(t) at time t, with no external forces acting, the equation simply

says

d2 x(t)

= 0.

m

dt 2

Note that this equation is indeed invariant under the Galilean group.

Emmy Noetherā™s principle of conserved quantities says that for a physical system

with a symmetry group, there are as many conserved quantities (constants of the

system unchanged as a function of time) as parameters for the group. As noted above,

the Galilean group depends on 10 parameters, so we are looking for 10 conserved

quantities. For a system with n particles having masses m i and position vectors xi (t),

Table 9.3 describes the conserved quantities of Newtonian dynamics.

Newtonian dynamics functioned well as a description of spacetime up until the

9.3.2

late nineteenth century. At that time however, two new developments shattered its

The

foundations. The ļ¬rst nail in its cofļ¬n was the famous Michelsonā“Morley experiment

PoincarĀ“ e

(1887), which refuted the best current explanation of the properties of light within

group and

Newtonian theory in terms of the ā˜theory of etherā™. The simplest interpretation of

special

their result was that the speed of light was independent of the speed of the observer,

relativity

in stark contradiction with the Galilean group, which obviously cannot accommodate

174 CONCLUDING REMARKS

such behaviour. A second (closely related) fact involves Maxwellā™s equations of

electromagnetism, which are not invariant under the Galilean group.

After an exciting decade of developments, best summarised elsewhere, Einsteinā™s

1905 foundational paper spelled out a new theory, special relativity, based on a

different set of principles. Four dimensional spacetime is henceforth to be mod-

elled on R1,3 , which is shorthand for a space with coordinates x = (t, x1 , x2 , x3 ) and

Lorentz pseudometric

ds 2 = ā’c2 dt 2 + dx1 + dx2 + dx3 ;

2 2 2

or, if the inļ¬nitesimal notation is unfamiliar, you can write the Lorentz distance of

vectors x = (t, xi ), y = (s, yi ) ā R1,3 as

d(x, y) = ā’c2 (t ā’ s)2 + (xi ā’ yi )2 .

i

(The sign we adopt is the opposite to most physics texts.) Here the constant c, with

the classical dimensions length/time, is the speed of light, postulated to be universal

in all inertial coordinate systems. In theoretical discussions, one often sets c = 1 for

reasons of convenience.

In special relativity, the only restriction on changes of reference frame is that the

Lorentz (pseudo-)distance on R1,3 (and the ā˜positive light-coneā™) is preserved; this

is Einsteinā™s relativity principle. The group of such transformations is the PoincarĀ“ e

group1 Poin(1, 3) consisting of maps

x ā’ Ax + b,

where A ā O+ (1, 3) is a Lorentz matrix (preserving the positive cone), and b ā R1,3 .

This group can be studied in complete analogy with the treatment of 6.5.3: it is the

semidirect product

Poin(1, 3) ā¼ O+ (1, 3) R1,3

=

of a normal subgroup, the group R1,3 of spacetime translations, and the four dimen-

sional Lorentz group O+ (1, 3). Also, for ļ¬xed values of the time variable t, the metric

reduces to the Euclidean metric on a copy of R3 . Hence Poin(1, 3) contains a subgroup

Eucl(3) of Euclidean transformations. However, since the PoincarĀ“ group mixes t and

e

x coordinates, this splitting of spacetime into ā˜timeā™ and ā˜spaceā™ is not canonical, but

depends on the choice of coordinate frame (observer).

Hyperbolic geometry is contained in the Lorentz space R1,n of special relativity

as the space-like hypersurface

q L (t, xi ) = ā’1 with t > 0.

1 The naming of concepts during these exciting years was rather haphazard, often respecting accident

and scientiļ¬c standing more than historical accuracy. In particular, the so-called Lorentz metric appears

to have been proposed ļ¬rst (albeit implicitly) by the Irish physicist George FitzGerald, followed (now

explicitly) by another Irishman, Sir Joseph Larmor and only for the third time by Lorentz himself. PoincarĀ“

e

came very close to inventing special relativity in the years 1900ā“1904, showing in particular that Lorentz

transformations form a group; hence in the case of the PoincarĀ“ group, the name is accurate.

e

9.3 GEOMETRY IN PHYSICS 175

The distinction of time-like and space-like vectors in the Lorentz model of hyperbolic

geometry derives exactly from this physical interpretation.

As discussed above, the PoincarĀ“ group Poin(1, 3) contains the Euclidean group

e

9.3.3

Eucl(3), hence also the Euclidean rotation group SO(3). As you recall from 8.5ā“8.6,

Wignerā™s

the latter group has a double cover SU(2) ā’ SO(3), that is, a two-to-one surjective

classifica-

group homomorphism with kernel Ā±1. It turns out that this double cover extends to

tion:

a double cover

elementary

particles

Poin(1, 3) ā’ Poin(1, 3)

of the PoincarĀ“ group, which can be constructed using the group SL(2, C) of

e

2 Ć— 2 complex matrixes of determinant 1 (which obviously contains the group SU(2)

covering SO(3)).

One of the ļ¬rst spectacular uses of group theory in theoretical physics was Wignerā™s

insight of the 1940s, which relates ā˜symmetries of spacetimeā™ to ā˜things in itā™ (parti-

cles), and can be summarised as follows (see Sternberg [23] for the physical intuition

and more details).

(1) An ā˜elementary particleā™ of nature is a (ļ¬nite dimensional, irreducible, unitary) rep-

resentation of the symmetry group of spacetime, satisfying certain ā˜physical restric-

tionsā™.

(2) The symmetry group of spacetime is the PoincarĀ“ group, or more precisely its universal

e

cover Poin(1, 3).

(3) The classiļ¬cation of the relevant representations of the PoincarĀ“ group thus leads to

e

a classiļ¬cation of all elementary particles.

Recall from 8.8 that a (linear) representation of a group G is a group homomorphism

from G to a group of (complex) matrixes; a unitary representation is one where the

image of every element of G is a unitary matrix (the latter restriction arises from

quantum mechanics, which need not unduly worry us at this point).

Wigner proved that ā˜physically relevantā™ representations of Poin(1, 3) are classiļ¬ed

by

r a continuous nonnegative parameter m ā„ 0, called the rest mass of the particle, and

r a half-integer s, called particle spin, that is allowed to take nonnegative values

0, 1 , 1, . . . for particles of mass m > 0, and all values 0, Ā± 1 , Ā±1, . . . for those with

2 2

m = 0.

Integral spin particles correspond to representations for which the kernel Ā±1 =

ker(Poin(1, 3) ā’ Poin(1, 3)) acts trivially, so really representations of Poin(1, 3);

whereas for particles with half-integral spin, the double covering is necessary.

Examples of the two kinds are photons, which are massless (that is, m = 0) and

have integral spin s = 1, and electrons with s = 1 and a certain positive value of m.

2

(The phenomenon of spin 1 particles was the main point of the discussion of 8.7.)

2

The group Poin(1, 3) has additional ā˜nonphysicalā™ representations with m 2 < 0; these

176 CONCLUDING REMARKS

are called tachyons (mythical particles travelling faster than the speed of light), and

are relegated to the world of science ļ¬ction in most current theories (but not all).

The importance of Wignerā™s insight in the development of modern physics can hardly

9.3.4

be overstated: in a sense, it concludes another 2000 plus year old story, the search for

The

the ultimate building blocks of the physical universe, and does so in mathematical

Standard

terms. Of Wignerā™s program, (1) and (3) have stood as cornerstones of most theories

Model and

of particle physics proposed in the last 50 years. Only (2), the speciļ¬c choice of the

beyond

symmetry group, has changed during the course of subsequent developments.

One thing that was clear already at the outset is that Wignerā™s original discussion

does not incorporate the electromagnetic interactions of elementary particles. This

however only requires a minor modiļ¬cation, taking into account an additional internal

symmetry group U(1). This group is no longer a geometric symmetry of spacetime,

but rather a symmetry of the whole theory of electromagnetism in spacetime, used to

encode additional data. Representations of the combined group Poin Ć— U(1) are now

parametrised by a triple of numbers (m, q, s), with the additional quantum number

q, the electric charge, taking integer values. In fact, internal symmetry groups such

as the U(1) of electromagnetism do not have to appear as a single group for the

whole theory; much more powerfully, each particle can have a ļ¬bre bundle of these

symmetry groups over the whole of spacetime, leading to the idea of gauge theory.

As the particle accelerators of the 1950s and 1960s grew capable of producing

faster and faster particles and slamming them into one another at higher and higher

energies, the zoo of known elementary particles grew accordingly. Alongside this, the

internal symmetry group also changed, accommodating various features of particles

to do with newly discovered forces, the strong and weak nuclear forces of particle

physics. In Wigner style, new groups led in turn to the prediction of new particles,

and their existence was in many cases conļ¬rmed in subsequent accelerator experi-

ments. There is really no space here to elaborate on this development; I recommend

Sternberg [23] as a good source. Let me only say that the most popular current theory

is the Standard Model, based on the PoincarĀ“ group augmented by the internal sym-

e

metry group U(1) Ć— SU(2) Ć— SU(3); roughly, the three factors are responsible for the

electromagnetic, weak and strong forces (this is of course a gross over-simpliļ¬cation).

Embedding the internal symmetry group U(1) Ć— SU(2) Ć— SU(3) into an even larger

group, mixing all three forces (electromagnetic, weak and strong) completely, come

under the name Grand Uniļ¬cation Theory (GUT), a sometime favourite pastime of

ā˜armchair physicsā™. Popular GUT groups include the special unitary group SU(5),

the group SO(10), and even more exotic constructs such as the ā˜exceptionalā™ groups

called E 6 and E 8 . It is hard, however, for any of these exotic theories to establish

a domination over their rivals; part of the problem seems to be that the Standard

Model works so well, and explains to remarkable accuracy almost everything one

could hope to see in experiments using accelerators of the present and near future;

thus anomalous measurements against which you can check your latest GUT group

are few and far between.

9.4 THE FAMOUS TRICHOTOMY 177

The connections between geometry and physics extend beyond the relationship

9.3.5

between spacetime symmetries and particles. The two crowning achievements

Other

of early twentieth century physics, quantum theory and general relativity, are

connections

inextricably linked to the ideas of geometry in a number of ways. The inļ¬‚uence

of the discovery of hyperbolic geometry on relativity has already been mentioned:

the fact that hyperbolic geometry has intrinsic curvature changed physical intuition,

culminating in Einsteinā™s insight that gravity, instead of acting as a classical ā˜forceā™,

is better described as encoded into the local curved structure of space itself (for more

on this, see the next section). Quantum mechanics, invented by SchrĀØ dinger and

o

Heisenberg in the 1920s, was axiomatised by Dirac and von Neumann, building on

the Hilbert incidence axioms for projective geometry (see 5.12). Much more recently,

the essential incompatibility between general relativity and quantum theory has led

to the introduction and study of string theory, which builds on and generalises all of

classical and modern geometry as we know it; this is however well beyond the scope of

this book.

9.4 The famous trichotomy

9.4.1 The metric geometries of this course come in a triad: spherical, Euclidean and hy-

The perbolic. In terms of curvature, the three geometries correspond to the three cases

curvature of Figure 9.4a, having local curvature positive, zero or negative. You can determine

trichotomy which geometry you are in locally by measuring the perimeter of a circle of radius

in geometry R, which, as you remember from Exercises 3.1 and 3.13, comes out to be 2Ļ sin R,

2Ļ R and 2Ļ sinh R in the three cases. The key point here is that the perimeter of a

circle or the area of a disc grows exponentially with the radius in hyperbolic space,

making hyperbolic space ā˜much biggerā™ than the sphere or the Euclidean plane. The

curvature can also be detected by measuring the angle sum of a triangle of the

geometry, which is > Ļ , equal to Ļ and < Ļ in the three cases, where the excess or

defect is proportional to the area of . Globally, as discussed at several points, the

difference is visible also in the incidence properties of lines: in the sphere two lines

always meet, in the Euclidean they either meet or are precisely parallel, whereas the

hyperbolic plane has plenty of pairs of lines that diverge.

Topologically, the Euclidean plane E2 , the sphere S 2 and hyperbolic space H2

are all simply connected (cf. 7.15; for H2 , use the homeomorphic model H of Exer-

cises 3.23ā“3.26 if you wish). As well as these simply connected geometries however,

we can also consider compact ones; for simplicity we only discuss the oriented sur-

faces here. The sphere is already compact; the compact version of the plane is the

one-holed torus, obtained from the plane by an equivalence relation which identiļ¬es

points which are related to each other by translation by vectors in a ļ¬xed parallelo-

gram lattice. The most exciting story is that of the hyperbolic plane, which by itself

can give rise to a multitude of compact geometric spaces: it can be shown that all

compact geometric surfaces with ā„ 2 holes can be derived from the hyperbolic plane

(Figure 9.4b). The number of holes in a compact surface is called its genus; so in

terms of the genus, our trichotomy becomes g = 0, g = 1 or g > 1. To return to the

178 CONCLUDING REMARKS

Figure 9.4a The cap, flat plane and Pringleā™s chip.

E2 H2

S2

...

g = 0, Ļ = 2

g > 1, Ļ < 0

g = 1, Ļ = 0

The genus trichotomy g = 0, g = 1, g ā„ 2 for oriented surfaces.

Figure 9.4b

basic trichotomy of positive, zero or negative curvature, we can take the Euler number

Ļ = 2 ā’ 2g of the surface, which is simply the quantity ā˜faces ā’ edges + vertexesā™

in Eulerā™s formula for a triangulated surface. Then Ļ = 2 for a sphere, as everyone

knows; also Ļ = 0 for a torus and Ļ < 0 for the geometric surfaces with more than

one hole. It is a fun exercise to triangulate a surface with two holes and check Eulerā™s

formula for it! (See Exercise 7.19 for the details.)

The classiļ¬cation of three dimensional geometries that extend our two dimensional

curvature trichotomy rejoices in the name of Thurstonā™s geometrisation conjecture

(late 1970s). This includes as a humble ļ¬rst case the PoincarĀ“ conjecture characterising

e

the 3-sphere; this may well turn out to be the ļ¬rst of the Clay Mathematical Instituteā™s

million-dollar Millennium Prize Problems to be solved. In a different direction, my

own subject of classiļ¬cation of varieties in algebraic geometry studies geometric

shapes deļ¬ned in space by several polynomial equations; the curvature trichotomy

reappears there in an algebraic form.

Much was written up to the turn of the twentieth century on the subject of whether

9.4.2

our own three dimensional universe is Euclidean, spherical or hyperbolic; PoincarĀ“ ā™s e

On the

extended essay La science et lā™hypoth` se (1902) points out that the question itself

e

shape and

begs a number of conventions, for example on how the objects of geometry (straight

fate of the

lines, distance) are realised as physical objects (light rays, observations of astronomy).

universe

Maybe the answer to the question depends on our choice of conventions.

9.4 THE FAMOUS TRICHOTOMY 179

The universe has grown in size and complexity since PoincarĀ“ ā™s day, an expansion

e

that continues apace to this day. According to special relativity (1905), it does not

make sense to consider space as a separate entity from spacetime. General relativity

(1916) says that spacetime is not ļ¬‚at or even of constant curvature, but is curved by

the presence of matter; this resolves the instantaneous action-at-a-distance that was a

philosophical contradiction implicit in Newtonā™s theory of gravitation. The existence

of black holes seems to be acknowledged by the majority of astrophysicists and

cosmologists, and the origin of the universe in the Big Bang some 13 Ć— 109 years ago

(give or take the odd billion years) is current orthodoxy. On a simple-minded view,

these extreme events of spacetime can only be represented in geometry as singularities

localised around isolated points. However, it is possible that the singularity is only

in our representation, much as Mercatorā™s projection presents a distorted view of the

North pole.

A separate trichotomy concerns the long-term future of the universe ā“ will gravity

eventually slow down the expansion of the universe, causing it to collapse back on

itself to a Big Crunch, so that time is also bounded in the future? will the expansion

continue indeļ¬nitely, with the universe getting bigger and bigger and emptier and

emptier? or are we precisely on the boundary between the two cases, so that expansion

slows down to nothing? The two trichotomies are possibly logically independent, but

who am I to judge?

One could believe that the general relativistic curvature effects of mass can be

envisaged as merely minor localised disturbances, and that space in the large is nev-

ertheless Euclidean; this is possibly the view held by many practising cosmologists

(I have not carried out a scientiļ¬c poll). However, it seems that the same population

cheerfully admits that something like 80ā“90% of the mass of the universe is not ac-

counted for by current theories (ā˜black matterā™ and ā˜black energyā™). Some will even

admit to not having any very specially well informed view on whether spacetime is

4-dimensional or really 10- or 11-dimensional. Just a little overall curvature or cos-

mological constant could go a long way (compare Exercise 3.13 (c)). Given all the

surprises that the study of science has brought to light in recent centuries, it might

seem premature to commit oneself to an excessively ļ¬rm view. There is a ļ¬‚ourishing

popular science literature on all these topics; perhaps the best informed books are

those of Martin Rees, for example [20].

Even if one admits the ļ¬‚at and boring possibility that the universe is asymptotically

9.4.3

Euclidean, and its expansion exactly ļ¬ne tuned to slow down but never reverse, it might

The snack

still happen that we get sucked into a black hole, and (who knows?) are resurrected

bar at the

to come out the other side as a new baby universe. At this point, you can pick and

end of the

choose what you want to believe, making this a nice optimistic note on which to end

universe

my fairy story.

Appendix A Metrics

A metric on a set X is a speciļ¬cation of a distance d(x, y) between

Definition

any two points x, y ā X , in other words a map d : X Ć— X ā’ R, required to satisfy

the following axioms for all x, y, z ā X :

d(x, y) ā„ 0 and d(x, y) = 0 if and only if x = y;

1.

d(x, y) = d(y, x);

2.

the triangle inequality d(x, y) ā¤ d(x, z) + d(z, y).

3.

For example, the real line R with d(x, y) = |x ā’ y| is a metric space. The epsilon-

delta deļ¬nition of continuity of a function in a ļ¬rst calculus course uses that R is a

metric space (compare 7.2). Theorem 1.1, Corollary 3.3 and Corollary 3.10 say that the

vector space Rn and hence Euclidean space En , the sphere S 2 and the hyperbolic plane

H2 are all metric spaces with their respective distance functions. The set of complex

numbers C is also a metric space under the distance function d(z 1 , z 2 ) = |z 2 ā’ z 1 |.

Some frivolous examples show that many distance functions in use in the real world

are not metrics:

1. Air fares: let d(x, y) be the price of an airline ticket from x to y; this is usually

unsymmetric, and does not satisfy the triangle inequality.

2. The distance you travel by car to go from one point of a town to another; this is not

symmetric, because of one-way trafļ¬c systems. However, it satisļ¬es the triangle in-

equality, because you take the minimum over paths, at least if your taxi driver is honest.

3. For a cyclist, up a hill is of course much further than down.

I use the following simple deļ¬nition to pass from a metric space to the slightly

more general notion of topological space in Chapter 7 (see Section 7.2).

Let X be a metric space, x ā X a point and Īµ > 0 a real number. The

Definition

ball in X of radius Īµ centred at x is the subset

B(x, Īµ) = y ā X d(x, y) < Īµ ā‚ X.

180

EXERCISES 181

For example, if X = R is the real line, then B(x, Īµ) is the usual open interval

(x ā’ Īµ, x + Īµ). All the deļ¬nitions of continuity of f (x) in the ļ¬rst calculus course

can be expressed in terms of these intervals.

Let (X, d) and (Y, dY ) be metric spaces. An isometry is a bijective

Definition

map f : X ā’ Y satisfying the condition

dY ( f (x), f (y)) = d(x, y).

The meaning of this deļ¬nition is that the two spaces (X, d) and (Y, dY ) are ā˜the

sameā™ as far as their metric properties are concerned. An example that is used very

often is the fact that the complex numbers C and the vector space R2 are isometric

under the map x + iy ā’ (x, y). Note that seemingly different metric spaces can be

isometric under some weird or ingenious map; see for example Exercise A.3 and, for

a geometric example, Exercise 3.24.

A slightly different case of this deļ¬nition that comes up all the time in geometry

is when (X, d) = (Y, d ) and f is a bijection. Then f is viewed as a selfmap of X

ā˜preserving all the metric geometryā™. The motions of geometries studied throughout

this book provide examples.

Exercises

Let X be a metric space and t : X ā’ X a map that preserves distances d(t(x), t(y)) =

A.1

d(x, y). Prove that t is injective. Give an example in which t is not bijective; in other

words, X can be isometric to a strict subset of itself, just as in set theory, an inļ¬nite

set can be in bijection with a strict subset. [Hint: think of ā˜Hilbertā™s hotelā™.]

Let S = [1, . . . , n] be a set containing n elements, and X the set of all subsets of S.

A.2

For x, y ā X , write d(x, y) for the size of the symmetric difference of x and y (the

number of elements of S contained in one of x, y but not the other). Show that d is a

metric on the set X . What happens to the construction if S is inļ¬nite? What happens

if S is inļ¬nite but I insist that X consists only of the ļ¬nite subsets of S?

Let P be the set of polynomials in one variable with coefļ¬cients in Z/2; remember,

A.3

this means that we work over the ļ¬eld {0, 1} with two elements where the addition

law includes 1 + 1 = 0. If f and g are two polynomials, let d( f, g) be the number

of nonzero terms in the difference f ā’ g. Show that d is a metric on P. Show also

that P with this metric is isometric to some metric space appearing in the previous

exercise.

Prove that a metric space with exactly 3 points is isometric to a subset of E2 .

A.4

Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check

A.5

that d is a metric. Prove that the metric space X is not isometric to any subset of En

for any n. Can you realise X as a subset of a sphere S 2 of appropriate radius, with

the spherical ā˜great circleā™ metric? [Hint: I am sure you know the riddle: an explorer

starts out from base camp, walks 10 miles due South, meets a bear, runs 10 miles due

West, then 10 miles due North and ļ¬nds himself back at base camp. What colour was

the bear? If in doubt, turn to Figure A.1.]

182 METRICS

Figure A.1 The bear.

Appendix B Linear algebra

The distance function in Rn is given by the norm |x|2 = xi2 , which comes from the

standard inner product x Ā· y = xi yi . The ideas here are familiar from Pythagorasā™

theorem and the equations of conics in plane geometry, and from the vector manipu-

lations in R3 used in applied math courses. A quadratic form in variables x1 , . . . , xn

is simply a homogeneous quadratic function in the obvious sense. For clarity I recall

the formal deļ¬nitions and results from linear algebra.

B.1 Bilinear form and quadratic form

Let V be a ļ¬nite dimensional vector space over R. A symmetric bi-

Definition

linear form Ļ• on V is a map Ļ• : V Ć— V ā’ R such that

Ļ• is linear in each of the two arguments, that is

(i)

Ļ•(Ī»u + Āµv, w) = Ī»Ļ•(u, w) + ĀµĻ•(v, w)

for all u, v, w ā V , Ī», Āµ ā R, and similarly for the second argument,

Ļ•(u, v) = Ļ•(v, u) for all u, v ā V.

(ii)

A quadratic form q on V is a map q : V ā’ R such that

q(Ī»u + Āµv) = Ī»2 q(u) + 2Ī»ĀµĻ•(u, v) + Āµ2 q(v)

for all u, v ā V , Ī», Āµ ā R, where Ļ•(u, v) is a symmetric bilinear form.

A quadratic form is determined by a symmetric bilinear form and

Proposition

vice versa by the rules

1

q(x) = Ļ•(x, x) and Ļ•(x, y) = q(x + y) ā’ q(x) ā’ q(y) .

2

183

184 LINEAR ALGEBRA

Choosing a basis e1 , . . . , en of V, a quadratic form q or its associated symmetric

bilinear form Ļ• are given by

q(x) = ai j xi x j = t xK x, Ļ•(x, y) = ai j xi y j = t xK y.

i, j i, j

Here x = t(x1 , . . . , xn ) = xi ei , y = t(y1 , . . . , yn ) = yi ei and K = (ki j ) is a

symmetric matrix whose entries are given by ki j = Ļ•(ei , e j ).

B.2 Euclid and Lorentz

There are two special bilinear forms that are useful in geometry. To see the ļ¬rst, let

V = Rn be the vector space with the standard basis

e1 = t(1, 0, . . . , 0), . . . , en = t(0, . . . , 0, 1).

The Euclidean inner product corresponds to the matrix

I = diag(1, 1, . . . , 1).

It is the familiar

Ļ• E (x, y) = x Ā· y = t xI y = xi yi ,

i

with corresponding quadratic form

q E (x) = |x|2 = xi2 .

i

As you know, an orthonormal basis of Rn is a set of n vectors f1 , . . . , fn ā Rn such

that

for i = j

0

fi Ā· f j = Ī“i j =

for i = j.

1

The model for this deļ¬nition is the usual basis ei = (0, . . . , 1, 0, . . . ) of Rn (with 1

in the ith place). The inner product Ļ• E expressed in terms of an orthonormal basis

f1 , . . . , fn of V still has matrix I.

For the indeļ¬nite case, it is convenient to change notation slightly, so let V = Rn+1

be the vector space with the standard basis e0 , . . . , en . The Lorentz dot product is the

symmetric bilinear form given by the matrix

J = diag(ā’1, 1, . . . , 1).

If x = (t, x1 , . . . , xn ) and y = (s, y1 , . . . , yn ) then

Ļ• L (x, y) = (t, x1 , . . . , xn ) Ā· L (s, y1 , . . . , yn ) = ā’ts + xi yi .

The Lorentz norm is the associated quadratic form q L : V ā’ R, deļ¬ned by

q L (t, x1 , . . . , xn ) = ā’t 2 + xi2 .

B.3 COMPLEMENTS AND BASES 185

A Lorentz basis f0 , f1 , . . . , fn is a basis of V as a vector space, with respect to which

q L has the standard diagonal matrix J ; that is,

q L (f0 ) = ā’1, q L (fi ) = 1 for i ā„ 1 fi Ā· L f j = 0 for i = j.

and

B.3 Complements and bases

Let (V, Ļ•) be a vector space with bilinear form.

For a vector subspace W ā‚ V , deļ¬ne the complement of W with

Definition

respect to Ļ• to be

Wā„ = x ā V Ļ•(x, w) = 0 for all w ā W .

In general, complements need not have any particularly nice properties; notice

for example that the zero inner product (with matrix K = 0) gives W ā„ = V for all

subspaces W . However, for ā˜niceā™ inner products the situation is completely different.

I write this section explicitly with the minimal generality needed for the geometric

applications; all this can be souped up to obtain the general Gramā“Schmidt process,

Sylvesterā™s law of inertia, etc.

Let Ļ• be the Euclidean inner product on V = Rn . Let W be a subspace

Theorem

of Rn . Then

W has an orthonormal basis f1 , . . . , fk ,

(1)

any vector v ā Rn has a unique expression v = w + u with w ā W and u ā W ā„ ; in

(2)

other words, Rn is the direct sum W ā• W ā„ .

Suppose that W is not the zero vector space, take a nonzero v1 ā V and let

Proof

f1 = v1 /|v1 | be a vector with unit length in the direction of v1 . If f1 spans W then I am

home. If not, take v2 outside the span of f1 and let f2 be a unit vector in the direction

of v2 ā’ (v2 Ā· f1 )f2 . Then, as you can check, the cunning choice of the direction of f2

ensures that it is orthogonal to f1 , and it lies in W . Now continue this way by induction.

Either the constructed f1 , . . . , fk generate W , or you can ļ¬nd vk+1 ā W outside their

span, and then a unit vector in the direction of vk+1 ā’ (vk+1 Ā· fi )fi can be added to

the collection.

For the second statement, ļ¬nd an orthonormal basis f1 , . . . , fk of W , and extend it

using the same method to an orthonormal basis f1 , . . . , fn of Rn . Then every vector

v ā Rn has a unique expression

n

v= Ī»i fi

i=1

and then

k n

w= Ī»i fi , u = Ī»i fi

i=1 i=k+1

is the only possible choice. QED

186 LINEAR ALGEBRA

The procedure of the proof is algorithmic, so lends itself easily to calculations; to

make sure that you understand it, do Exercise B.1.

Let V = Rn+1 with the Lorentz dot product and form.

Theorem

Let v ā Rn+1 be any vector with q L (v) < 0. Then q L (w, w) > 0 for w a nonzero vector

(3)

in the Lorentz complement vā„ .

Let f0 ā Rn+1 be a vector with q L (f0 ) = ā’1. Then f0 is part of a Lorentz basis f0 , . . . , fn

(4)

of Rn+1 .

For (3), suppose that v = (t, x1 , . . . , xn ) and w = (s, y1 , . . . , yn ) satisfy

Proof

q L (v) < 0 and v Ā· L w = 0, that is

n

ā’t + xi2 < 0

2

(1)

i=1

and

n

ā’st + xi yi = 0. (2)

i=1

Then (1) and (2) give that

n n

ā’s + t = ā’s t + t

2

yi2 2 22 2

yi2

i=1 i=1

n n n

2

>ā’ + yi2 ,

xi2

xi yi

i=1 i=1 i=1

provided that the yi are not all 0. But we know that the last line is ā„ 0 (in fact it is

equal to (xi y j ā’ x j yi )2 , compare 1.1), so

n

ā’s + yi2 > 0

2

i=1

which is the statement.

For (4), pick v1 ā Rn+1 linearly independent of f0 and set

w1 = v1 + (f0 Ā· L v1 )f0 .

Then w1 is a nonzero element of fā„ , so by (3) it has positive Lorentz norm. Hence I

ā 0

can set f1 = v1 / q L (v1 ). Then by construction f0 , f1 are part of a Lorentz basis. Now

continue with the inductive method used in the proof of the previous theorem. QED

B.4 Symmetries

Return to the case of a general symmetric bilinear form Ļ• on the vector space V , and

its associated quadratic form q.

B.5 ORTHOGONAL AND LORENTZ MATRIXES 187

Let Ī± : V ā’ V be a linear map. Then equivalent conditions:

Proposition

Ī± preserves q, that is, q(Ī±(x)) = q(x) for all x ā V ,

1.

Ī± preserves Ļ•, that is, Ļ•(Ī±(x), Ī±(y)) = Ļ•(x, y) for all x, y ā V .

2.

The equivalence simply follows from the fact that q is determined by Ļ•

Proof

and conversely, Ļ• is determined by q from Proposition B.1. QED

Now identify V with Rn using the standard basis e1 , . . . , en . Let K = {Ļ•(ei , e j )}

be the matrix of Ļ•.

Let A be the n Ć— n matrix representing Ī± in the given

Proposition (continued)

basis. Then the previous two conditions are also equivalent to

A satisļ¬es the matrix equality tAK A = K .

3.

Recall Ļ•(x, y) = t xAy. Hence

Proof

Ļ•(Ī±(x), Ī±(y)) = Ļ•(x, y) āā’ t(Ax)K (Ay) = t xtAK Ay = t xK y

and the latter holds for all x and y if and only if tAK A = K . QED

A useful observation is the following.

If det K = 0 (we say that the form Ļ• is nondegenerate) then the equivalent

Lemma

conditions above imply det A = Ā±1.

From (3) and properties of the determinant it follows that

Proof

(det A)2 det K = det K .

If det K = 0 then I can divide by it. QED

B.5 Orthogonal and Lorentz matrixes

Consider Rn with the Euclidean inner product, and let e1 , . . . , en with ei =

(0, . . . , 1, 0, . . . ) be the usual basis. If f1 , . . . , fn ā Rn are any n vectors, there is

a unique linear map Ī± : Rn ā’ Rn such that Ī±(ei ) = fi for i = 1, . . . , n. Namely

write f j as the column vector f j = (ai j ); then Ī± is given by the matrix A = (ai j )

with columns the vectors f j . Now, by Proposition B.4 and by direct inspection, the

following conditions are equivalent:

f1 , . . . , fn is an orthonormal basis;

1.

2. the columns of A form an orthonormal basis;

AA = I;

t

3.

Ī± preserves the Euclidean inner product.

4.

We say that Ī± is an orthogonal transformation and A an orthogonal matrix if these

conditions hold. We get the following result.

188 LINEAR ALGEBRA

Ī± ā’ (Ī±(e1 ), . . . , Ī±(en )) establishes a one-to-one correspondence

Proposition

orthogonal transformations orthonormal bases

ā” .

Ī± of R f 1 , . . . , fn ā R n

n

If (V, Ļ•) is Lorentz, a matrix A satisfying the condition tA J A = J of Propo-

sition B.4 (3) is called a Lorentz matrix. I leave you to formulate the analogous

correspondence between Lorentz bases and Lorentz matrixes.

B.6 Hermitian forms and unitary matrixes

This section discusses a slight variant of the above material, for vector spaces over

the ļ¬eld C of complex numbers. Let V be a ļ¬nite dimensional vector space over C.

A Hermitian form Ļ• : V Ć— V ā’ C is a map satisfying the conditions

Ļ•(Ī»u + Āµv, w) = Ī»Ļ•(u, w) + ĀµĻ•(v, w)

and

Ļ•(u, Ī»v + Āµw) = Ī»Ļ•(u, v) + ĀµĻ•(u, w),

where Ī», Āµ ā C; note the appearance of the complex conjugate in the ļ¬rst row. The

corresponding Hermitian norm q on V is

q(v) = Ļ•(v, v).

The relation between Ļ• and q is slightly more complicated than in the real case; I

leave you to check the rather daunting looking identity

1

Ļ•(u, v) = q(u + v) ā’ q(u ā’ v) + iq(u + iv) ā’ iq(u ā’ iv) .

4

The terms in the identity are not so important; what is important is the fact that q

gives back Ļ•.

Since I am only interested in a special case, I choose a basis {e1 , . . . , en } of V

straight away and assume that

Ļ•(Ī»1 e1 + Ā· Ā· Ā· + Ī»n en , Āµ1 e1 + Ā· Ā· Ā· + Āµn en ) = Ī»1 Āµ1 + Ā· Ā· Ā· + Ī»n Āµn .

Such a form is called a deļ¬nite Hermitian form. Under Ļ•, e1 , . . . , en form a Hermitian

or orthonormal basis: Ļ•(ei , e j ) = Ī“i j .

The following is completely analogous to Proposition B.4.

Let Ī± : V ā’ V be a linear map represented by the n Ć— n matrix A

Proposition

in the given basis. Then the following are equivalent:

Ī± preserves the norm q;

1.

Ī± preserves the Hermitian form Ļ•;

2.

EXERCISES 189

A satisļ¬es hA A = In , where hA is the Hermitian conjugate deļ¬ned by hA = tA; that

3.

is, (hA)i j = A ji .

The transformation Ī± or the matrix A representing it is unitary if it satisļ¬es these

conditions; the set of n Ć— n unitary matrixes is denoted U(n). Unitary transformations

(possibly on inļ¬nite dimensional spaces) have many pleasant properties which makes

them ubiquitous in mathematics. They are also the basic building blocks of quantum

mechanics and hence presumably nature; in this book I discuss one tiny example of

this in 8.7.

Exercises

Let f1 = (2/3, 1/3, 2/3) and f2 = (1/3, 2/3, ā’2/3) ā R3 ; ļ¬nd all vectors f3 ā R3 for

B.1

which f1 , f2 , f3 is an orthonormal basis.

By writing down explicitly the conditions for a 2 Ć— 2 matrix to be Lorentz, show that

B.2

any such matrix has the form

ā’ sinh s

cosh s sinh s cosh s

.

or

ā’ cosh s

sinh s cosh s sinh s

B.3 This exercise is a generalisation of the previous one; it shows that any Lorentz matrix

can be put in a simple normal form in a suitable Lorentz basis; the Euclidean case is

included in the main text in 1.11. Let Ī± : Rn+1 ā’ Rn+1 be a linear map given by a

Lorentz matrix A. Prove that there exists a Lorentz basis of Rn+1 in which the matrix

of Ī± is

ļ£« Ā±1 ļ£¶ ļ£« B0 ļ£¶

Ik + Ik +

ļ£¬ ļ£· ļ£¬ ļ£·

ļ£¬ ļ£· ļ£¬ ļ£·

ā’Ik ā’ ā’Ik ā’

B=ļ£¬ ļ£· B=ļ£¬ ļ£·

or

ļ£¬ ļ£· ļ£¬ ļ£·

B1 B1

ļ£ ļ£ø ļ£ ļ£ø

.. ..

. .

Bl Blā’1

where B0 = Ā± cosh Īø00 cosh Īø00 , Bi = cos Īøii ā’ sinĪøĪøi for i > 0, and Ik Ā± are identity ma-

Īø

Īø sinh

Īø

sinh sin cos i

trixes. [Hint: argue as in the Euclidean case in 1.11.2; the only extra complication is

that you have to take into account the sign of the Lorentz form on the eigenvectors.

The statement follows by sorting out the cases that can arise.]

Prove that a unitary matrix has determinant det A ā C of absolute value 1.

B.4

References

[1] Michael Artin, Algebra, Englewood Cliffs, NJ: Prentice Hall, 1991.

[2] Alan F. Beardon, The Geometry of Discrete Groups, New York: Springer, 1983.

[3] Roberto Bonola, Non-Euclidean Geometry: a Critical and Historical Study of its Developments,

New York: Dover, 1955.

[4] J. H. Conway and D. A. Smith, On Quaternions and Octonions, Natick, MA: A. K. Peters,

2002.

[5] H. S. M. Coxeter, Introduction to Geometry, 2nd edn, New York: Wiley, 1969.

[6] Peter H. Dana, The Geographerā™s Craft Project 1999, http://www.colorado.edu/geography/

gcraft/notes/mapproj/mapproj.html.

[7] Richard P. Feynman, The Feynman Lectures on Physics, Vol. 3: Quantum Mechanics, Reading,

MA: Addison-Wesley, 1965.

[8] C. M. R. Fowler, The Solid Earth, Cambridge: Cambridge University Press, 1990.

[9] William Fulton and Joseph Harris, Representation Theory, a First Course, Readings in Math-

ematics, New York: Springer, 1991.

[10] James A. Green, Sets and Groups, a First Course in Algebra, London: Chapman and Hall,

1995.

[11] Marvin J. Greenberg, Euclidean and non-Euclidean Geometries: Development and History,

3rd edn, New York: W. H. Freeman, 1993.

[12] Robin Hartshorne, Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, New

York: Springer, 2000.

[13] David Hilbert, Foundations of Geometry, 2nd edn, LaSalle: Open Court, 1971.

[14] Walter Ledermann, Introduction to the Theory of Finite Groups, Edinburgh: Oliver and Boyd,

1964.

[15] Pertti Lounesto, Clifford Algebras and Spinors, Cambridge: Cambridge University Press, 1997.

[16] Dana Mackenzie, A sine on the road to Mecca, American Scientist, 89 (3) (Mayā“June 2001).

[17] P. M. Neumann, G. A. Story and E. C. Thompson, Groups and Geometry, Oxford: Oxford

University Press, 1994.

[18] V. V. Nikulin and I. R. Shafarevich, Geometries and Groups, Berlin: Springer Universitext,

1987.

[19] Elmer Rees, Notes on Geometry, Berlin: Springer, 1983.

[20] Martin Rees, Before the Beginning, Simon and Schuster, 1997.

[21] Walter Rudin, Principles of Mathematical Analysis, 3rd edn, New York: McGraw-Hill,

1976.

190

REFERENCES 191

[22] Graeme Segal, Lie groups, in R. Carter, G. Segal and I. G. Macdonald, Lectures on Lie groups

and Lie algebras, CUP/LMS student texts, Cambridge: Cambridge University Press, 1995.

[23] Shlomo Sternberg, Group Theory and Physics, Cambridge: Cambridge University Press, 1994.

[24] W. A. Sutherland, Introduction to Metric and Topological Spaces, Oxford: Clarendon Press,

1975.

Index

and bounded, 115ā“129, 146

abstract group, 169

diagonal, 127ā“128

afļ¬ne

map, 129ā“130

frame, 69, 71

coļ¬nite topology, 108, 111, 127

geometry, 62ā“72, 95

commutative law, 15, 17, 28, 32

group Aff(n), 102, 161, 170

compact, see maximal ā“, sequentially ā“, xv, 75,

linear

115ā“117, 121, 133ā“138, 143, 146,

dependence, 68, 71

152

map, 8ā“9, 27, 68ā“69

Lie group, 146, 147, 160

subspace, 29ā“30, 62ā“68, 70ā“72, 91

space An , 62, 63, 68, 95, 170 surface, 119, 177ā“178

versus closed, 128ā“129

in projective space, 82

compactiļ¬cation, 75

span, 62, 66ā“67

complex number, 12, 27, 136, 188

transformation, xvi, 8ā“9, 68ā“70, 91

composite

algebraic topology, xv, 113, 130

of maps, 26ā“33

algebraically closed ļ¬eld, 136ā“137

of reļ¬‚ections, 16, 29ā“31, 33, 58

angle, 1, 5ā“6, 27, 62, 69, 95

of rotation and glide, 33

bisector, 23, 25

of rotation and reļ¬‚ection, 31

of rotation, 15ā“18

of rotations, 27, 33

signed, 6

of translations, 27

sum, 19ā“20, 34, 40, 51ā“56

congruent triangles, 19, 25, 55

angular

connected, see path ā“, simply ā“, 113ā“115, 117,

defect, excess, see angle sum

138, 148, 149, 152, 153

momentum, 93, 154

component, 114ā“115, 122, 144, 148, 149, 153,

area, 40ā“41, 51ā“56

160, 161

associative law, 28, 32, 94, 169

Lie group, 160

axiomatic projective geometry, 86ā“88, 164, 168,

continuous, xv, 5, 68, 91, 100, 142ā“144, 148,

177

149

family of paths, 131ā“132

ball, 58, 109, 138, 146

contractible loop, 130ā“133, 136,

based loop, 131ā“133, 136ā“137

141

basis for a topology, 124ā“126

coordinate

bilinear form, see Euclidean inner product,

changes, xiv

Lorentz dot product, 162, 183ā“185

frame, xiv, 1

Bolyaiā™s letter, 166

geometry, xiii, xvi, 168

system, xiv, 4

centre of rotation, 15

Coventry market, 92ā“93

centroid, 21, 69ā“71

cross-ratio, 79ā“81, 90, 106

circumcentre, 21, 22

curvature, 34, 40, 49, 93, 167, 177, 178,

closed, see compact versus closed, 58, 75, 108,

182

111, 113, 138, 148

193

194 INDEX

Desarguesā™ theorem, 82ā“84, 88, 90 group, see abstract ā“, fundamental ā“, Galilean ā“,

dimension, 66, 67, 70, 76, 144, 145, 160 general linear ā“, Lie ā“, Lorentz ā“,

of a Lie group, 144ā“146, 148, 161 PoincarĀ“ ā“, projective linear ā“,

e

of intersection, 67, 69, 72ā“73, 77, 81, 83, 88 reļ¬‚ection ā“, rotation ā“, spinor ā“,

direct motion, 10, 15, 17, 148, 151ā“152 topological ā“, transformation ā“, unitary ā“

disc, 111, 122, 130, 133, 139

discrete topology, 108, 110, 127, 143 half-turn, 12, 32

distance, see Euclidean ā“, hyperbolic ā“, metric, Hausdorff, 109, 110, 127ā“130, 139, 152

shortest ā“, spherical ā“ Heineā“Borel theorem, 116

function, 1, 2, 4, 6, 7, 35, 62, 95, 180, 181, 183 Hermitian form, 153, 156, 160, 163, 188

duality, 85ā“86, 90 homeomorphism, 107, 111, 113, 117, 119ā“121,

130, 132, 134ā“136, 138, 139, 147, 149,

Einsteinā™s 152, 153, 160, 177

ļ¬eld equations, see general relativity, 93 criterion, 111, 130, 142, 152

relativity principle, see special relativity, 174 problem, xv, 113

electron, xvi, 143, 154ā“159, 175 homogeneous space, 169ā“170

empty set, 68, 70, 72, 73, 76, 108, 124 hyperbolic

Erlangen program, xivā“xv, 95ā“96, 112, 170ā“171 distance, 43, 46, 58

Euclidā™s postulates, see parallel postulate, geometry, 4, 20, 34, 36, 41ā“167

165ā“167 line, 43, 46ā“50, 60

Euclidean motion, 46, 144

plane H2 , 39, 47ā“49, 58ā“61, 180

angle, 45

distance, 1, 2, 4, 116, 151 sine rule, 59

frame, 1, 14, 25, 40, 145 space, 35, 42, 51, 104

geometry, 4, 19, 25, 34, 45, 47, 69, 95, 166 translation, 47, 58, 61

group Eucl(n), xvi, 159, 161 triangle, 44, 51, 58, 59

inner product, 2, 5, 9, 24, 43, 58, 184, 185, 187 trig, 35, 44ā“45

line, 4 hyperplane, 29, 30, 66, 67, 76, 78, 81, 82, 89, 96

motion, see motion, 9, 10, 14, 24, 25, 47, 92, at inļ¬nity, 88

144

plane E2 , 6, 33 ideal point, see inļ¬nity, point at

space En , 1, 4ā“10, 29, 35, 180 ideal triangle, 51, 53ā“56

translation, 19 incentre, 23, 25

Euler number, 140, 177 incidence of lines, 34, 40, 47, 69, 84

indiscrete topology, 108, 111, 139

family of paths, 131 inļ¬nity

Feuerbach circle, 23 hyperplane at, 72, 73, 76, 82, 90

frame, see afļ¬ne ā“, coordinate ā“, Euclidean ā“, point at, 48ā“49, 51, 53, 55, 59, 73, 75, 76,

orthogonal ā“, projective ā“, spherical ā“ 79

frame of reference, see projective frame intersection, see dimension of ā“, 108

fundamental intrinsic

group, xv, 113, 130, 159 curvature, 34, 40, 177

theorem of algebra, 136 distance, 40

unit, 34, 49

Galilean group, 93, 172ā“173 isometry, see motion, preserves distances, 4, 6,

general 112, 181

linear group GL(n), xv, 95, 99, 101, 105, 124,

143, 145, 147, 148, 160, 161, 171 Klein bottle, xiv, 139

relativity, 93, 167, 176, 178

generators, 29, 100ā“101, 103, 106 length of path, 5

genus, 120, 139, 177 Lie group, see compact ā“, 142ā“164, 169

geodesic, see shortest distance line, 4, 65

glide, 15ā“17, 24, 31ā“33, 40, 47, 98 hyperbolic, 44

reļ¬‚ection, see glide segment, 3, 65

glueing, see quotient topology spherical, 35

great circle, see spherical line loop, 107ā“137, 140, 159

INDEX 195

Lorentz axes, 31

basis, 44, 55, 185, 186, 188, 189 hyperplanes, 17, 64, 66, 67

complement, 48, 186 lines, 15ā“17, 20ā“23, 27, 34, 40, 49, 62, 68, 70,

dot product Ā· L , 43, 184, 186 73, 82, 166

form q L , 42, 47, 184, 186, 189 mirrors, 103

group, 93, 159, 161 postulate, 20, 49, 60, 166

matrix, 42, 46ā“47, 161, 188, 189 sides, 31

norm, 44, 184, 186 vector, 16, 96

orthogonal, 44 path, see length of path, minimum over paths, 114,

matrix, 187 131, 159

pseudometric, 42, 58, 174 connected, 114, 120, 132, 141, 149

reļ¬‚ection, 47 perpendicular bisector, 16, 21, 22, 24, 29, 30,

space, 42, 46, 188 57

transformation, 47, 54, 92, 144 perspective, 73, 74, 81ā“83, 88, 90

translation, see hyperbolic ā“ physics, xv, xvi, 93, 160, 172ā“179

PoincarĀ“ group, 173ā“176

e

maximal compact subgroup, 160 point at inļ¬nity, see inļ¬nity, point at

Mercatorā™s projection, 139, 164, 179 preserves distances, 6ā“7, 24, 39, 181

metric, 180ā“182 principal homogeneous space, see torsor

geometry, 64, 177 Pringleā™s potato chip, 58, 178

space, 1, 4, 38, 180ā“182 product topology, 126ā“127, 139, 143

topology, 109, 125, 143, 152 proļ¬nite topology, 125, 126

minimum over paths, 5, 180 projective

MĀØ bius strip, xiv, 107, 118ā“119, 122, 139

o frame, 78, 79, 90, 106, 146

motion, xiv, 1, 6, 7, 9ā“11, 14ā“19, 24ā“26, 28ā“34, geometry, 72ā“91

38ā“40, 46, 47, 58, 61, 93, 95, 97, 98, 100, linear group PGL(n), 77, 95, 105, 106, 144,

103, 105, 106, 144, 149, 151, 152, 154, 146, 171

158, 161 linear subspace, 73ā“77

punctured disc D ā— , 120, 130, 133, 136

mousetrap topology, 122ā“123

MusĀ“ e GrĀ“ vin, 103, 105

e e

quadratic form, 5, 9, 42, 123, 150, 151, 183

Newtonian dynamics, 93, 161, 172ā“173 quaternions, 149ā“152

non-Euclidean geometry, 34ā“61, 167 quotient topology, 110, 117ā“119, 121ā“125,

normal form of a matrix, 10ā“13, 18, 29, 98ā“99, 139ā“140, 144, 152

148, 189

reļ¬‚ection, 1, 11, 15ā“17, 24, 27ā“30, 33, 34, 40, 58,

open set, 108ā“111, 113ā“115, 117, 118, 121, 125, 103, 105

143, 148 group, 103ā“105

opposite motion, 10, 15, 17, 148 matrix, 7, 10, 24, 42, 144

orthocentre, 22ā“23 relativity, see special ā“, general ā“, 161

orthogonal, see Lorentz ā“ rigid body motion, see motion

axes, 1 rotary reļ¬‚ection, 33, 40

complement V ā„ , 13, 47, 145, 171, 185 rotation, 1, 11, 15ā“18, 24, 25, 27, 29, 31ā“34, 39,

direct sum, 151 40, 47, 97, 100, 103, 142, 143, 149ā“152,

frame, 39 154, 158, 161

group O(n), 144ā“152 group, 152

line, 158 matrix, 7, 10, 42, 144

magnetic ļ¬eld, 154, 158 rubber-sheet geometry, xiv, 107

matrix, 7, 9ā“13, 24, 29, 39, 99, 144, 146ā“149,

159, 187 sequentially compact, 115ā“116, 138

plane, 29 shortest distance, see minimum over paths, 4, 5,

transformation, 9, 92, 99, 187 40, 46, 58

vector, 5, 29, 37, 151, 162, 185 similar triangles, 21ā“23

simplex of reference, see projective frame

Pappusā™ theorem, 84ā“85, 88, 90 simply connected, 130, 132, 146, 160

parallel spacetime, 93, 172ā“176, 178, 179

196 INDEX

special topology, 94, 107ā“141, 143

of Pn , 90, 121, 139

linear group SL(n), 159, 175

orthogonal group SO(n), 149, 152 of SO(3), 142, 143, 149

of S 3 , 152

relativity, xv, 93, 144, 173ā“174, 178

unitary group SU(n), 153, 176 torsor, 169ā“170

sphere S 2 , 35, 36, 39, 40, 43, 56, 58, 113, 180, 181 torus, 119, 120, 139, 177, 178

sphere S n , 57, 58, 116, 121, 122, 145, 151 transformation group, 26ā“33, 92, 94ā“96, 101, 104,

spherical 112, 142ā“163

disc, 56 translation, 1, 15ā“19, 25, 29, 31ā“33, 39, 68, 97,

distance, 36ā“38, 40, 56, 116 98, 100ā“103, 106, 158, 161

frame, 34, 40 map, 125

geometry, 4, 20, 34ā“41, 45, 56, 57, 164, 167, subgroup, 101, 105

182 vector, 15, 24, 27, 31

line, 39, 40 triangle inequality, 1ā“5, 38, 45, 180

motion, 38, 39 trichotomy, 177ā“179

triangle, 37ā“38, 40, 41, 57, 182

trig, 37, 167 ultraparallel lines, 48ā“51, 59, 61

spin, 143, 154, 155 UMP, see universal mapping property

spinor group Spin(n), 153, 159 unitary

Standard Model, 176 group, 153, 176

subspace topology, 117, 121, 128, 144, 147, 152 matrix, 153, 158, 188ā“189

symmetry, 92ā“95, 160, 164, 169, 173ā“176 representation, 175

universal mapping property, 118, 139,

topological 152

group, 143ā“144, 159

property, xv, 113, 127, 131, 136, 167 winding number, xv, 107, 130ā“137

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