<< стр. 8(всего 8)СОДЕРЖАНИЕ
homo-
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 afп¬Ѓne 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
afп¬Ѓne geometry does not have an invariant distance function.
If is a G-set and x в€€ , the stabiliser subgroup of x is the set of elements of G
that п¬Ѓx x, that is

StabG (x) = h в€€ G h(x) = x .

For example, the stabiliser subgroup of the origin 0 в€€ En in Eucl(n) is the group O(n)
of orthogonal matrixes.
If G acts transitively on , the map ex : G в†’ deп¬Ѓned by g в†’ gx is surjective.
Moreover elements g1 , g2 в€€ G map to the same point of if and only if g2 = g1 h
for some h в€€ StabG (x); thus ex induces a bijection G/ StabG (x) в€’ . (Here G/H
в†’
stands for the quotient of G by the equivalence relation g в€ј gh for h в€€ 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 в€€ . 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 в€€ .

For example, afп¬Ѓne space An is a homogeneous space under Aff(n), but is a torsor
under the translation subgroup Rn вЉ‚ Aff(n).
According to the previous discussion, if is a G-torsor, then ex : G в†’ 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 вЉ‚
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 в€€ is of order two, consisting of the identity and the reп¬‚ection in the axis through
P. The subgroup H acts transitively and without stabilisers (since it does not contain
reп¬‚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 KleinвЂ™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. HelmholtzвЂ™ 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 в€€ En is isomorphic to the group O(n), the group of rotations
and reп¬‚ections п¬Ѓxing 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 в€€ 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 в€€ 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 deп¬Ѓnition of afп¬Ѓne space I gave in Chapter 4.
9.2.4
Let V be a vector space; an afп¬Ѓne space A(V ) is just a torsor under V . In other words,
Affine space
A(V ) is a set with an action of V (вЂ˜by translationвЂ™), and this action is simply transitive:
as a torsor
for P, Q в€€ V there is a unique vector x в€€ 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 afп¬Ѓne 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 deп¬Ѓne Euclidean space to be the
space of all conjugates of a п¬Ѓxed copy of O(n) вЉ‚ Eucl(n), and notions of Euclidean
geometry to be all notions that can be deп¬Ѓned 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 afп¬Ѓne transformations Aff(V )
(see 4.5). This contains copies of GL(V ), the group of invertible linear maps of V , as
afп¬Ѓne transformations п¬Ѓxing 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 П•v : P в†’
P + v of the additive group of V deп¬Ѓned on points of afп¬Ѓne space. For P в€€ A(V ),
we get a bijection e P : V в†’ A(V ) mapping v в€€ V to P + v; two such identiп¬Ѓca-
tions differ by an element of V acting by translation. The bijections e P are differ-
ent coordinate systems on afп¬Ѓne 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
в€’в†’
P, Q в€€ A(V ) determine a vector e P (Q) = P Q в€€ 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 п¬Ѓrst 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-simpliп¬Ѓed; 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 Г— 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 в€’ 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) в†’ (Ax + gt + b, t + s),

where A в€€ O(3) is a 3 Г— 3 orthogonal matrix, g and b are 3 Г— 1 column vectors, and
s в€€ R is a scalar. Such transformations collectively form the Galilean group Gal(3, 1)
of classical (3 + 1)-dimensional spacetime E3 Г— 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) в†’ (x + b, t) i
dxi
m i xi Г—
spatial rotation angular momentum
dt
(x, t) в†’ (Ax, t) i

в€’pt +
Galilean boost m i xi centre of mass (where
(x, t) в†’ (x + gt, t) i p is the total momen-
tum)
1 dxi 2
time translation mi energy
2 dt
(x, t) в†’ (x, t + s) i

(x, t) в†’ (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 п¬Ѓxed 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 NewtonвЂ™s equation of motion, in the
form familiar to you from a п¬Ѓrst 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 NoetherвЂ™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 п¬Ѓrst nail in its cofп¬Ѓn was the famous MichelsonвЂ“Morley experiment
PoincarВґ e
(1887), which refuted the best current explanation of the properties of light within
group and
Newtonian theory in terms of the вЂ˜theory of etherвЂ™. 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 MaxwellвЂ™s equations of
electromagnetism, which are not invariant under the Galilean group.
After an exciting decade of developments, best summarised elsewhere, EinsteinвЂ™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 = в€’c2 dt 2 + dx1 + dx2 + dx3 ;
2 2 2

or, if the inп¬Ѓnitesimal notation is unfamiliar, you can write the Lorentz distance of
vectors x = (t, xi ), y = (s, yi ) в€€ R1,3 as

d(x, y) = в€’c2 (t в€’ s)2 + (xi в€’ 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 вЂ˜positive light-coneвЂ™) is preserved; this
is EinsteinвЂ™s relativity principle. The group of such transformations is the PoincarВґ e
group1 Poin(1, 3) consisting of maps

x в†’ Ax + b,

where A в€€ O+ (1, 3) is a Lorentz matrix (preserving the positive cone), and b в€€ 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) в€ј 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 п¬Ѓxed 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 PoincarВґ group mixes t and
e
x coordinates, this splitting of spacetime into вЂ˜timeвЂ™ and вЂ˜spaceвЂ™ 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 ) = в€’1 with t > 0.
1 The naming of concepts during these exciting years was rather haphazard, often respecting accident
and scientiп¬Ѓc standing more than historical accuracy. In particular, the so-called Lorentz metric appears
to have been proposed п¬Ѓrst (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. PoincarВґ
e
came very close to inventing special relativity in the years 1900вЂ“1904, showing in particular that Lorentz
transformations form a group; hence in the case of the PoincarВґ 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 PoincarВґ 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 8.5вЂ“8.6,
WignerвЂ™s
the latter group has a double cover SU(2) в†’ SO(3), that is, a two-to-one surjective
classifica-
group homomorphism with kernel В±1. It turns out that this double cover extends to
tion:
a double cover
elementary
particles
Poin(1, 3) в†’ Poin(1, 3)

of the PoincarВґ group, which can be constructed using the group SL(2, C) of
e
2 Г— 2 complex matrixes of determinant 1 (which obviously contains the group SU(2)
covering SO(3)).
One of the п¬Ѓrst spectacular uses of group theory in theoretical physics was WignerвЂ™s
insight of the 1940s, which relates вЂ˜symmetries of spacetimeвЂ™ to вЂ˜things in itвЂ™ (parti-
cles), and can be summarised as follows (see Sternberg  for the physical intuition
and more details).

(1) An вЂ˜elementary particleвЂ™ of nature is a (п¬Ѓnite dimensional, irreducible, unitary) rep-
resentation of the symmetry group of spacetime, satisfying certain вЂ˜physical restric-
tionsвЂ™.
(2) The symmetry group of spacetime is the PoincarВґ group, or more precisely its universal
e
cover Poin(1, 3).
(3) The classiп¬Ѓcation of the relevant representations of the PoincarВґ group thus leads to
e
a classiп¬Ѓcation 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 вЂ˜physically relevantвЂ™ representations of Poin(1, 3) are classiп¬Ѓed
by
r a continuous nonnegative parameter m в‰Ґ 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, В± 1 , В±1, . . . for those with
2 2
m = 0.

Integral spin particles correspond to representations for which the kernel В±1 =
ker(Poin(1, 3) в†’ 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 вЂ˜nonphysicalвЂ™ 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 п¬Ѓction in most current theories (but not all).

The importance of WignerвЂ™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 WignerвЂ™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 speciп¬Ѓc 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 WignerвЂ™s original discussion
does not incorporate the electromagnetic interactions of elementary particles. This
however only requires a minor modiп¬Ѓcation, 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 Г— 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 п¬Ѓbre 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 conп¬Ѓrmed in subsequent accelerator experi-
ments. There is really no space here to elaborate on this development; I recommend
Sternberg  as a good source. Let me only say that the most popular current theory
is the Standard Model, based on the PoincarВґ group augmented by the internal sym-
e
metry group U(1) Г— SU(2) Г— SU(3); roughly, the three factors are responsible for the
electromagnetic, weak and strong forces (this is of course a gross over-simpliп¬Ѓcation).
Embedding the internal symmetry group U(1) Г— SU(2) Г— SU(3) into an even larger
group, mixing all three forces (electromagnetic, weak and strong) completely, come
under the name Grand Uniп¬Ѓcation Theory (GUT), a sometime favourite pastime of
вЂ˜armchair physicsвЂ™. Popular GUT groups include the special unitary group SU(5),
the group SO(10), and even more exotic constructs such as the вЂ˜exceptionalвЂ™ 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 inп¬‚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 EinsteinвЂ™s insight that gravity, instead of acting as a classical вЂ˜forceвЂ™,
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 SchrВЁ 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 2ПЂ sin R,
2ПЂ R and 2ПЂ 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 вЂ˜much biggerвЂ™ 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 > ПЂ , equal to ПЂ and < ПЂ 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 3.23вЂ“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 identiп¬Ѓes
points which are related to each other by translation by vectors in a п¬Ѓxed 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 в‰Ґ 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 PringleвЂ™s chip.

E2 H2

S2

...
g = 0, П‡ = 2

g > 1, П‡ < 0
g = 1, П‡ = 0

The genus trichotomy g = 0, g = 1, g в‰Ґ 2 for oriented surfaces.
Figure 9.4b

basic trichotomy of positive, zero or negative curvature, we can take the Euler number
П‡ = 2 в€’ 2g of the surface, which is simply the quantity вЂ˜faces в€’ edges + vertexesвЂ™
in EulerвЂ™s formula for a triangulated surface. Then П‡ = 2 for a sphere, as everyone
knows; also П‡ = 0 for a torus and П‡ < 0 for the geometric surfaces with more than
one hole. It is a fun exercise to triangulate a surface with two holes and check EulerвЂ™s
formula for it! (See Exercise 7.19 for the details.)
The classiп¬Ѓcation of three dimensional geometries that extend our two dimensional
curvature trichotomy rejoices in the name of ThurstonвЂ™s geometrisation conjecture
(late 1970s). This includes as a humble п¬Ѓrst case the PoincarВґ conjecture characterising
e
the 3-sphere; this may well turn out to be the п¬Ѓrst of the Clay Mathematical InstituteвЂ™s
million-dollar Millennium Prize Problems to be solved. In a different direction, my
own subject of classiп¬Ѓcation of varieties in algebraic geometry studies geometric
shapes deп¬Ѓned 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; PoincarВґ вЂ™s e
On the
extended essay La science et lвЂ™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 PoincarВґ вЂ™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 п¬‚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 NewtonвЂ™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 Г— 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 MercatorвЂ™s projection presents a distorted view of the
North pole.
A separate trichotomy concerns the long-term future of the universe вЂ“ 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 indeп¬Ѓnitely, 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 scientiп¬Ѓc poll). However, it seems that the same population
cheerfully admits that something like 80вЂ“90% of the mass of the universe is not ac-
counted for by current theories (вЂ˜black matterвЂ™ and вЂ˜black energyвЂ™). 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 п¬Ѓrm view. There is a п¬‚ourishing
popular science literature on all these topics; perhaps the best informed books are
those of Martin Rees, for example .

Even if one admits the п¬‚at and boring possibility that the universe is asymptotically
9.4.3
Euclidean, and its expansion exactly п¬Ѓne 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 speciп¬Ѓcation of a distance d(x, y) between
Definition
any two points x, y в€€ X , in other words a map d : X Г— X в†’ R, required to satisfy
the following axioms for all x, y, z в€€ X :
d(x, y) в‰Ґ 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) в‰¤ d(x, z) + d(z, y).
3.

For example, the real line R with d(x, y) = |x в€’ y| is a metric space. The epsilon-
delta deп¬Ѓnition of continuity of a function in a п¬Ѓrst 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 в€’ 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 trafп¬Ѓc systems. However, it satisп¬Ѓes 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 deп¬Ѓnition 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 в€€ X a point and Оµ > 0 a real number. The
Definition
ball in X of radius Оµ centred at x is the subset

B(x, Оµ) = y в€€ X d(x, y) < Оµ вЉ‚ X.

180
EXERCISES 181

For example, if X = R is the real line, then B(x, Оµ) is the usual open interval
(x в€’ Оµ, x + Оµ). All the deп¬Ѓnitions of continuity of f (x) in the п¬Ѓrst 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 в†’ Y satisfying the condition
dY ( f (x), f (y)) = d(x, y).
The meaning of this deп¬Ѓnition is that the two spaces (X, d) and (Y, dY ) are вЂ˜the
sameвЂ™ 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 в†’ (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 deп¬Ѓnition 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
вЂ˜preserving all the metric geometryвЂ™. The motions of geometries studied throughout
this book provide examples.

Exercises
Let X be a metric space and t : X в†’ 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 inп¬Ѓnite
set can be in bijection with a strict subset. [Hint: think of вЂ˜HilbertвЂ™s hotelвЂ™.]
Let S = [1, . . . , n] be a set containing n elements, and X the set of all subsets of S.
A.2
For x, y в€€ 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 inп¬Ѓnite? What happens
if S is inп¬Ѓnite but I insist that X consists only of the п¬Ѓnite subsets of S?
Let P be the set of polynomials in one variable with coefп¬Ѓcients in Z/2; remember,
A.3
this means that we work over the п¬Ѓeld {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 в€’ 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 вЂ˜great circleвЂ™ 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 п¬Ѓnds 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 В· y = xi yi . The ideas here are familiar from PythagorasвЂ™
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 deп¬Ѓnitions and results from linear algebra.

B.1 Bilinear form and quadratic form
Let V be a п¬Ѓnite dimensional vector space over R. A symmetric bi-
Definition
linear form П• on V is a map П• : V Г— V в†’ R such that

П• is linear in each of the two arguments, that is
(i)

П•(О»u + Вµv, w) = О»П•(u, w) + ВµП•(v, w)

for all u, v, w в€€ V , О», Вµ в€€ R, and similarly for the second argument,
П•(u, v) = П•(v, u) for all u, v в€€ V.
(ii)

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

q(О»u + Вµv) = О»2 q(u) + 2О»ВµП•(u, v) + Вµ2 q(v)

for all u, v в€€ V , О», Вµ в€€ R, where П•(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) = П•(x, x) and П•(x, y) = q(x + y) в€’ q(x) в€’ q(y) .
2

183
184 LINEAR ALGEBRA

Choosing a basis e1 , . . . , en of V, a quadratic form q or its associated symmetric
bilinear form П• are given by

q(x) = ai j xi x j = t xK x, П•(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 = П•(ei , e j ).

B.2 Euclid and Lorentz
There are two special bilinear forms that are useful in geometry. To see the п¬Ѓrst, 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
П• E (x, y) = x В· y = t xI y = xi yi ,
i

q E (x) = |x|2 = xi2 .
i
As you know, an orthonormal basis of Rn is a set of n vectors f1 , . . . , fn в€€ Rn such
that

for i = j
0
fi В· f j = Оґi j =
for i = j.
1
The model for this deп¬Ѓnition is the usual basis ei = (0, . . . , 1, 0, . . . ) of Rn (with 1
in the ith place). The inner product П• E expressed in terms of an orthonormal basis
f1 , . . . , fn of V still has matrix I.
For the indeп¬Ѓnite 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(в€’1, 1, . . . , 1).
If x = (t, x1 , . . . , xn ) and y = (s, y1 , . . . , yn ) then

П• L (x, y) = (t, x1 , . . . , xn ) В· L (s, y1 , . . . , yn ) = в€’ts + xi yi .

The Lorentz norm is the associated quadratic form q L : V в†’ R, deп¬Ѓned by

q L (t, x1 , . . . , xn ) = в€’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 ) = в€’1, q L (fi ) = 1 for i в‰Ґ 1 fi В· L f j = 0 for i = j.
and

B.3 Complements and bases
Let (V, П•) be a vector space with bilinear form.

For a vector subspace W вЉ‚ V , deп¬Ѓne the complement of W with
Definition
respect to П• to be

WвЉҐ = x в€€ V П•(x, w) = 0 for all w в€€ 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 вЉҐ = V for all
subspaces W . However, for вЂ˜niceвЂ™ 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 GramвЂ“Schmidt process,
SylvesterвЂ™s law of inertia, etc.

Let П• 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 в€€ Rn has a unique expression v = w + u with w в€€ W and u в€€ W вЉҐ ; in
(2)
other words, Rn is the direct sum W вЉ• W вЉҐ .

Suppose that W is not the zero vector space, take a nonzero v1 в€€ 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 в€’ (v2 В· 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 п¬Ѓnd vk+1 в€€ W outside their
span, and then a unit vector in the direction of vk+1 в€’ (vk+1 В· fi )fi can be added to
the collection.
For the second statement, п¬Ѓnd 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 в€€ Rn has a unique expression
n
v= О»i fi
i=1
and then
k n
w= О»i fi , u = О»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 в€€ 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 vвЉҐ .
Let f0 в€€ Rn+1 be a vector with q L (f0 ) = в€’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 В· L w = 0, that is
n
в€’t + xi2 < 0
2
(1)
i=1

and
n
в€’st + xi yi = 0. (2)
i=1

Then (1) and (2) give that
n n
в€’s + t = в€’s t + t
2
yi2 2 22 2
yi2
i=1 i=1
n n n
2
>в€’ + 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 в‰Ґ 0 (in fact it is
equal to (xi y j в€’ x j yi )2 , compare 1.1), so
n
в€’s + yi2 > 0
2

i=1
which is the statement.
For (4), pick v1 в€€ Rn+1 linearly independent of f0 and set

w1 = v1 + (f0 В· L v1 )f0 .

Then w1 is a nonzero element of fвЉҐ , so by (3) it has positive Lorentz norm. Hence I
в€љ 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 П• on the vector space V , and
B.5 ORTHOGONAL AND LORENTZ MATRIXES 187

Let О± : V в†’ V be a linear map. Then equivalent conditions:
Proposition

О± preserves q, that is, q(О±(x)) = q(x) for all x в€€ V ,
1.
О± preserves П•, that is, П•(О±(x), О±(y)) = П•(x, y) for all x, y в€€ V .
2.

The equivalence simply follows from the fact that q is determined by П•
Proof
and conversely, П• is determined by q from Proposition B.1. QED
Now identify V with Rn using the standard basis e1 , . . . , en . Let K = {П•(ei , e j )}
be the matrix of П•.

Let A be the n Г— n matrix representing О± in the given
Proposition (continued)
basis. Then the previous two conditions are also equivalent to
A satisп¬Ѓes the matrix equality tAK A = K .
3.

Recall П•(x, y) = t xAy. Hence
Proof

П•(О±(x), О±(y)) = П•(x, y) в‡ђв‡’ 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 П• is nondegenerate) then the equivalent
Lemma
conditions above imply det A = В±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 в€€ Rn are any n vectors, there is
a unique linear map О± : Rn в†’ Rn such that О±(ei ) = fi for i = 1, . . . , n. Namely
write f j as the column vector f j = (ai j ); then О± 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.
О± preserves the Euclidean inner product.
4.
We say that О± is an orthogonal transformation and A an orthogonal matrix if these
conditions hold. We get the following result.
188 LINEAR ALGEBRA

О± в†’ (О±(e1 ), . . . , О±(en )) establishes a one-to-one correspondence
Proposition

orthogonal transformations orthonormal bases
в†” .
О± of R f 1 , . . . , fn в€€ R n
n

If (V, П•) 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 п¬Ѓeld C of complex numbers. Let V be a п¬Ѓnite dimensional vector space over C.
A Hermitian form П• : V Г— V в†’ C is a map satisfying the conditions

П•(О»u + Вµv, w) = О»П•(u, w) + ВµП•(v, w)

and

П•(u, О»v + Вµw) = О»П•(u, v) + ВµП•(u, w),

where О», Вµ в€€ C; note the appearance of the complex conjugate in the п¬Ѓrst row. The
corresponding Hermitian norm q on V is

q(v) = П•(v, v).

The relation between П• and q is slightly more complicated than in the real case; I
leave you to check the rather daunting looking identity

1
П•(u, v) = q(u + v) в€’ q(u в€’ v) + iq(u + iv) в€’ iq(u в€’ iv) .
4
The terms in the identity are not so important; what is important is the fact that q
gives back П•.
Since I am only interested in a special case, I choose a basis {e1 , . . . , en } of V
straight away and assume that

П•(О»1 e1 + В· В· В· + О»n en , Вµ1 e1 + В· В· В· + Вµn en ) = О»1 Вµ1 + В· В· В· + О»n Вµn .

Such a form is called a deп¬Ѓnite Hermitian form. Under П•, e1 , . . . , en form a Hermitian
or orthonormal basis: П•(ei , e j ) = Оґi j .
The following is completely analogous to Proposition B.4.

Let О± : V в†’ V be a linear map represented by the n Г— n matrix A
Proposition
in the given basis. Then the following are equivalent:
О± preserves the norm q;
1.
О± preserves the Hermitian form П•;
2.
EXERCISES 189

A satisп¬Ѓes hA A = In , where hA is the Hermitian conjugate deп¬Ѓned by hA = tA; that
3.
is, (hA)i j = A ji .

The transformation О± or the matrix A representing it is unitary if it satisп¬Ѓes these
conditions; the set of n Г— n unitary matrixes is denoted U(n). Unitary transformations
(possibly on inп¬Ѓnite 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, в€’2/3) в€€ R3 ; п¬Ѓnd all vectors f3 в€€ R3 for
B.1
which f1 , f2 , f3 is an orthonormal basis.
By writing down explicitly the conditions for a 2 Г— 2 matrix to be Lorentz, show that
B.2
any such matrix has the form

в€’ sinh s
cosh s sinh s cosh s
.
or
в€’ 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 О± : Rn+1 в†’ 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 О± is
пЈ« В±1 пЈ¶ пЈ« B0 пЈ¶
Ik + Ik +
пЈ¬ пЈ· пЈ¬ пЈ·
пЈ¬ пЈ· пЈ¬ пЈ·
в€’Ik в€’ в€’Ik в€’
B=пЈ¬ пЈ· B=пЈ¬ пЈ·
or
пЈ¬ пЈ· пЈ¬ пЈ·
B1 B1
пЈ­ пЈё пЈ­ пЈё
.. ..
. .
Bl Blв€’1

where B0 = В± cosh Оё00 cosh Оё00 , Bi = cos Оёii в€’ sinОёОёi for i > 0, and Ik В± are identity ma-
Оё
Оё sinh
Оё
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 в€€ C of absolute value 1.
B.4
References

 Michael Artin, Algebra, Englewood Cliffs, NJ: Prentice Hall, 1991.
 Alan F. Beardon, The Geometry of Discrete Groups, New York: Springer, 1983.
 Roberto Bonola, Non-Euclidean Geometry: a Critical and Historical Study of its Developments,
New York: Dover, 1955.
 J. H. Conway and D. A. Smith, On Quaternions and Octonions, Natick, MA: A. K. Peters,
2002.
 H. S. M. Coxeter, Introduction to Geometry, 2nd edn, New York: Wiley, 1969.
 Peter H. Dana, The GeographerвЂ™s Craft Project 1999, http://www.colorado.edu/geography/
gcraft/notes/mapproj/mapproj.html.
 Richard P. Feynman, The Feynman Lectures on Physics, Vol. 3: Quantum Mechanics, Reading,
 C. M. R. Fowler, The Solid Earth, Cambridge: Cambridge University Press, 1990.
 William Fulton and Joseph Harris, Representation Theory, a First Course, Readings in Math-
ematics, New York: Springer, 1991.
 James A. Green, Sets and Groups, a First Course in Algebra, London: Chapman and Hall,
1995.
 Marvin J. Greenberg, Euclidean and non-Euclidean Geometries: Development and History,
3rd edn, New York: W. H. Freeman, 1993.
 Robin Hartshorne, Geometry: Euclid and Beyond, Undergraduate Texts in Mathematics, New
York: Springer, 2000.
 David Hilbert, Foundations of Geometry, 2nd edn, LaSalle: Open Court, 1971.
 Walter Ledermann, Introduction to the Theory of Finite Groups, Edinburgh: Oliver and Boyd,
1964.
 Pertti Lounesto, Clifford Algebras and Spinors, Cambridge: Cambridge University Press, 1997.
 Dana Mackenzie, A sine on the road to Mecca, American Scientist, 89 (3) (MayвЂ“June 2001).
 P. M. Neumann, G. A. Story and E. C. Thompson, Groups and Geometry, Oxford: Oxford
University Press, 1994.
 V. V. Nikulin and I. R. Shafarevich, Geometries and Groups, Berlin: Springer Universitext,
1987.
 Elmer Rees, Notes on Geometry, Berlin: Springer, 1983.
 Martin Rees, Before the Beginning, Simon and Schuster, 1997.
 Walter Rudin, Principles of Mathematical Analysis, 3rd edn, New York: McGraw-Hill,
1976.

190
REFERENCES 191

 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.
 Shlomo Sternberg, Group Theory and Physics, Cambridge: Cambridge University Press, 1994.
 W. A. Sutherland, Introduction to Metric and Topological Spaces, Oxford: Clarendon Press,
1975.
Index

and bounded, 115вЂ“129, 146
abstract group, 169
diagonal, 127вЂ“128
afп¬Ѓne
map, 129вЂ“130
frame, 69, 71
coп¬Ѓnite topology, 108, 111, 127
geometry, 62вЂ“72, 95
commutative law, 15, 17, 28, 32
group Aff(n), 102, 161, 170
compact, see maximal вЂ“, sequentially вЂ“, xv, 75,
linear
115вЂ“117, 121, 133вЂ“138, 143, 146,
dependence, 68, 71
152
map, 8вЂ“9, 27, 68вЂ“69
Lie group, 146, 147, 160
subspace, 29вЂ“30, 62вЂ“68, 70вЂ“72, 91
space An , 62, 63, 68, 95, 170 surface, 119, 177вЂ“178
versus closed, 128вЂ“129
in projective space, 82
compactiп¬Ѓcation, 75
span, 62, 66вЂ“67
complex number, 12, 27, 136, 188
transformation, xvi, 8вЂ“9, 68вЂ“70, 91
composite
algebraic topology, xv, 113, 130
of maps, 26вЂ“33
algebraically closed п¬Ѓeld, 136вЂ“137
of reп¬‚ections, 16, 29вЂ“31, 33, 58
angle, 1, 5вЂ“6, 27, 62, 69, 95
of rotation and glide, 33
bisector, 23, 25
of rotation and reп¬‚ection, 31
of rotation, 15вЂ“18
of rotations, 27, 33
signed, 6
of translations, 27
sum, 19вЂ“20, 34, 40, 51вЂ“56
congruent triangles, 19, 25, 55
angular
connected, see path вЂ“, simply вЂ“, 113вЂ“115, 117,
defect, excess, see angle sum
138, 148, 149, 152, 153
momentum, 93, 154
component, 114вЂ“115, 122, 144, 148, 149, 153,
area, 40вЂ“41, 51вЂ“56
160, 161
associative law, 28, 32, 94, 169
Lie group, 160
axiomatic projective geometry, 86вЂ“88, 164, 168,
continuous, xv, 5, 68, 91, 100, 142вЂ“144, 148,
177
149
family of paths, 131вЂ“132
ball, 58, 109, 138, 146
contractible loop, 130вЂ“133, 136,
based loop, 131вЂ“133, 136вЂ“137
141
basis for a topology, 124вЂ“126
coordinate
bilinear form, see Euclidean inner product,
changes, xiv
Lorentz dot product, 162, 183вЂ“185
frame, xiv, 1
BolyaiвЂ™s letter, 166
geometry, xiii, xvi, 168
system, xiv, 4
centre of rotation, 15
Coventry market, 92вЂ“93
centroid, 21, 69вЂ“71
cross-ratio, 79вЂ“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

DesarguesвЂ™ theorem, 82вЂ“84, 88, 90 group, see abstract вЂ“, fundamental вЂ“, Galilean вЂ“,
dimension, 66, 67, 70, 76, 144, 145, 160 general linear вЂ“, Lie вЂ“, Lorentz вЂ“,
of a Lie group, 144вЂ“146, 148, 161 PoincarВґ вЂ“, projective linear вЂ“,
e
of intersection, 67, 69, 72вЂ“73, 77, 81, 83, 88 reп¬‚ection вЂ“, rotation вЂ“, spinor вЂ“,
direct motion, 10, 15, 17, 148, 151вЂ“152 topological вЂ“, transformation вЂ“, unitary вЂ“
disc, 111, 122, 130, 133, 139
discrete topology, 108, 110, 127, 143 half-turn, 12, 32
distance, see Euclidean вЂ“, hyperbolic вЂ“, metric, Hausdorff, 109, 110, 127вЂ“130, 139, 152
shortest вЂ“, spherical вЂ“ HeineвЂ“Borel theorem, 116
function, 1, 2, 4, 6, 7, 35, 62, 95, 180, 181, 183 Hermitian form, 153, 156, 160, 163, 188
duality, 85вЂ“86, 90 homeomorphism, 107, 111, 113, 117, 119вЂ“121,
130, 132, 134вЂ“136, 138, 139, 147, 149,
EinsteinвЂ™s 152, 153, 160, 177
п¬Ѓeld equations, see general relativity, 93 criterion, 111, 130, 142, 152
relativity principle, see special relativity, 174 problem, xv, 113
electron, xvi, 143, 154вЂ“159, 175 homogeneous space, 169вЂ“170
empty set, 68, 70, 72, 73, 76, 108, 124 hyperbolic
Erlangen program, xivвЂ“xv, 95вЂ“96, 112, 170вЂ“171 distance, 43, 46, 58
EuclidвЂ™s postulates, see parallel postulate, geometry, 4, 20, 34, 36, 41вЂ“167
165вЂ“167 line, 43, 46вЂ“50, 60
Euclidean motion, 46, 144
plane H2 , 39, 47вЂ“49, 58вЂ“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, 44вЂ“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 inп¬Ѓnity, 88
144
plane E2 , 6, 33 ideal point, see inп¬Ѓnity, point at
space En , 1, 4вЂ“10, 29, 35, 180 ideal triangle, 51, 53вЂ“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 inп¬Ѓnity
Feuerbach circle, 23 hyperplane at, 72, 73, 76, 82, 90
frame, see afп¬Ѓne вЂ“, coordinate вЂ“, Euclidean вЂ“, point at, 48вЂ“49, 51, 53, 55, 59, 73, 75, 76,
orthogonal вЂ“, projective вЂ“, spherical вЂ“ 79
frame of reference, see projective frame intersection, see dimension of вЂ“, 108
fundamental intrinsic
group, xv, 113, 130, 159 curvature, 34, 40, 177
theorem of algebra, 136 distance, 40
unit, 34, 49
Galilean group, 93, 172вЂ“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, 100вЂ“101, 103, 106 length of path, 5
genus, 120, 139, 177 Lie group, see compact вЂ“, 142вЂ“164, 169
geodesic, see shortest distance line, 4, 65
glide, 15вЂ“17, 24, 31вЂ“33, 40, 47, 98 hyperbolic, 44
reп¬‚ection, see glide segment, 3, 65
glueing, see quotient topology spherical, 35
great circle, see spherical line loop, 107вЂ“137, 140, 159
INDEX 195

Lorentz axes, 31
basis, 44, 55, 185, 186, 188, 189 hyperplanes, 17, 64, 66, 67
complement, 48, 186 lines, 15вЂ“17, 20вЂ“23, 27, 34, 40, 49, 62, 68, 70,
dot product В· 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, 46вЂ“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
reп¬‚ection, 47 perpendicular bisector, 16, 21, 22, 24, 29, 30,
space, 42, 46, 188 57
transformation, 47, 54, 92, 144 perspective, 73, 74, 81вЂ“83, 88, 90
translation, see hyperbolic вЂ“ physics, xv, xvi, 93, 160, 172вЂ“179
PoincarВґ group, 173вЂ“176
e
maximal compact subgroup, 160 point at inп¬Ѓnity, see inп¬Ѓnity, point at
MercatorвЂ™s projection, 139, 164, 179 preserves distances, 6вЂ“7, 24, 39, 181
metric, 180вЂ“182 principal homogeneous space, see torsor
geometry, 64, 177 PringleвЂ™s potato chip, 58, 178
space, 1, 4, 38, 180вЂ“182 product topology, 126вЂ“127, 139, 143
topology, 109, 125, 143, 152 proп¬Ѓnite topology, 125, 126
minimum over paths, 5, 180 projective
MВЁ bius strip, xiv, 107, 118вЂ“119, 122, 139
o frame, 78, 79, 90, 106, 146
motion, xiv, 1, 6, 7, 9вЂ“11, 14вЂ“19, 24вЂ“26, 28вЂ“34, geometry, 72вЂ“91
38вЂ“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, 73вЂ“77
punctured disc D в€— , 120, 130, 133, 136
mousetrap topology, 122вЂ“123
MusВґ e GrВґ vin, 103, 105
e e
quadratic form, 5, 9, 42, 123, 150, 151, 183
Newtonian dynamics, 93, 161, 172вЂ“173 quaternions, 149вЂ“152
non-Euclidean geometry, 34вЂ“61, 167 quotient topology, 110, 117вЂ“119, 121вЂ“125,
normal form of a matrix, 10вЂ“13, 18, 29, 98вЂ“99, 139вЂ“140, 144, 152
148, 189
reп¬‚ection, 1, 11, 15вЂ“17, 24, 27вЂ“30, 33, 34, 40, 58,
open set, 108вЂ“111, 113вЂ“115, 117, 118, 121, 125, 103, 105
143, 148 group, 103вЂ“105
opposite motion, 10, 15, 17, 148 matrix, 7, 10, 24, 42, 144
orthocentre, 22вЂ“23 relativity, see special вЂ“, general вЂ“, 161
orthogonal, see Lorentz вЂ“ rigid body motion, see motion
axes, 1 rotary reп¬‚ection, 33, 40
complement V вЉҐ , 13, 47, 145, 171, 185 rotation, 1, 11, 15вЂ“18, 24, 25, 27, 29, 31вЂ“34, 39,
direct sum, 151 40, 47, 97, 100, 103, 142, 143, 149вЂ“152,
frame, 39 154, 158, 161
group O(n), 144вЂ“152 group, 152
line, 158 matrix, 7, 10, 42, 144
magnetic п¬Ѓeld, 154, 158 rubber-sheet geometry, xiv, 107
matrix, 7, 9вЂ“13, 24, 29, 39, 99, 144, 146вЂ“149,
159, 187 sequentially compact, 115вЂ“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, 21вЂ“23
simplex of reference, see projective frame
PappusвЂ™ theorem, 84вЂ“85, 88, 90 simply connected, 130, 132, 146, 160
parallel spacetime, 93, 172вЂ“176, 178, 179
196 INDEX

special topology, 94, 107вЂ“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, 173вЂ“174, 178
unitary group SU(n), 153, 176 torsor, 169вЂ“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, 26вЂ“33, 92, 94вЂ“96, 101, 104,
spherical 112, 142вЂ“163
disc, 56 translation, 1, 15вЂ“19, 25, 29, 31вЂ“33, 39, 68, 97,
distance, 36вЂ“38, 40, 56, 116 98, 100вЂ“103, 106, 158, 161
frame, 34, 40 map, 125
geometry, 4, 20, 34вЂ“41, 45, 56, 57, 164, 167, subgroup, 101, 105
182 vector, 15, 24, 27, 31
line, 39, 40 triangle inequality, 1вЂ“5, 38, 45, 180
motion, 38, 39 trichotomy, 177вЂ“179
triangle, 37вЂ“38, 40, 41, 57, 182
trig, 37, 167 ultraparallel lines, 48вЂ“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, 188вЂ“189
symmetry, 92вЂ“95, 160, 164, 169, 173вЂ“176 representation, 175
universal mapping property, 118, 139,
topological 152
group, 143вЂ“144, 159
property, xv, 113, 127, 131, 136, 167 winding number, xv, 107, 130вЂ“137

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