\chapter{Concluding remarks}
\markboth{\quad\thechapter\ Concluding remarks\hfill}{}

This final chapter is quite different from the earlier ones in style and
intention: I let my hair down with a number of informal fairy stories on
different topics, tying together loose strands in the historical and
mathematical argument of the book, and opening up some new directions. In
particular, I give a ``popular science'' discussion of some of the
surprising and amazingly fertile links between the geometry, topology and
Lie group theory discussed in this book and different aspects of 20th
century physics. There are many more topics I would have liked to write
about, but life is short, so I compromise by allowing the discussion to
degenerate to a final list of bullet points. However, incompleteness is
intentional throughout, and I leave you to explore details from other
source, in or out of the confines of a degree course and its attendant
examinations.

\section{On the history of geometry}\label{sec!history}\label{sec!parallels}
\markright{\hfill\thesection\ History of geometry\quad}

\subsection{Greek geometry and rigour}

Geometry has a very special place in the history and culture of western
mathematics. Coming at the dawn of western civilisation ($350\pm200$ BC),
Greek philosophy and geometry, passed on to us by the more advanced culture
of the Islamic world at the time of the Renaissance, has played a central
role in the development of western culture, not merely for its content, but
for its idea of rigour. The Greeks were not the first to attempt to describe
the world around them by ``geometry'': that credit must go to the ancient
Mesopotamians (from 2,500 BC), followed by the Egyptians (from 2,000 BC).
However, before the Greeks, geometry largely consisted of a bag of tricks
for calculation that worked in practice most of the time. In contrast,
Greek mathematicians elaborated the notion of logical argument. By this I
do not mean the elementary and often hairsplitting logic of a
``Foundations'' or ``Set theory'' or ``Abstract algebra'' course, but the
idea that understanding steps at different stages in an argument from the
ground up is at least as important as somehow getting an approximately
correct answer. This is one of the fundamental items of intellectual
equipment that set Western mathematics and science apart from (and in the
course of time well above) that of India and China.

Building on sources largely unknown to us, the geometer Euclid, probably
working in Alexandria in the 4th century BC, summarised the mathematical
knowledge of the time in his 13 volume {\em Elements}. Book I deals with the
basic definitions of geometry. Euclid introduces notions such as {\em
point}, {\em line}, {\em plane}, {\em distance}, {\em angle} and {\em
meets}, whose meaning is supposed to be self-evident, and enunciates certain
{\em postulates} (in modern language, axioms) concerning these notions.
Lengths and angles are to be thought of as geometric quantities in their own
right, not related to any algebraic or numeric representation. For example,
one of the postulates states that two line segments are equal if they are
congruent, which makes perfect sense without having to consider the length
of a line as a number.

\subsection{The parallel postulate} \label{sec!parallelpostulate}

Most of Euclid's postulates were for a long time beyond doubt, but the last
one stood out from the beginning as far less obvious:
 \begin{quote}
If a line falls on two lines, with interior angles on one side adding to
less than two right angles, the two lines, if extended indefinitely, meet on
the side on which the angles add to less than two right angles.
\end{quote}
\begin{figure}[ht]
 \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}a}
 \centering \input{final_tomeet.pstex_t}
 \caption{The parallel postulate. To meet or not to meet, that is the
 question.}
 \label{tomeet}
 \end{figure}
This is nonobvious. Behold Figure~\ref{tomeet}! The ``extended
indefinitely'' of Euclid's phrase makes it clear that the statement involves
arguing on objects that are arbitrarily distant, so that it is in principle
not verifiable. Through the ages, many alternatives were formulated, which
can be proved to be equivalent to the above on the basis of the other
axioms, such as:
\begin{quote}
\index{parallel!postulate}\index{parallel!lines}
 given a line $L$ in the plane, and a point $P$ not on $L$, there exists one
 and only one line through $P$ not meeting~$L$
 \end{quote}
(compare Figure~\ref{parallel_axiom} and Figure~\ref{fig!hpll}). Or
\begin{quote}
\index{parallel!postulate}\index{parallel!lines}
the sum of the angles of a triangle is equal to two right angles
 \end{quote}
(see Theorem~\ref{angles_in_triangle} and Theorem~\ref{sec!ang.def}).

 \begin{figure}[ht]
 \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}b}
 \centering \input{intro_para_post.pstex_t}
 \caption{The parallel postulate in the Euclidean plane}
 \label{parallel_axiom}
 \end{figure}

After arguably the longest dispute in intellectual history, it was
discovered between about 1810 and 1830 by Bolyai, Gauss, Lobachevsky and
Schweikart (independently, alphabetical order) that the parallel postulate
cannot be a consequence of Euclid's other axioms: there exist axiomatic
geometries which are in many ways similar to Euclidean plane
geometry,\index{Euclidean!geometry} sharing its aesthetic appeal and
simplicity, but which do not satisfy the parallel postulate. As J\'anos
Bolyai wrote to his father,
\begin{quote}
``ollyan fels\'eges dolgokat hoztam ki, hogy magam elb\'amultam, s
\"or\"ok\"os k\'ar volna elveszni; ha megl\'atja \'Edes Ap\'am megesm\'eri;
most t\"obbet nem sz\'olhatok, tsak annyit: hogy semmib\H ol egy ujj m\'as
vil\'agot teremtettem; mind az, valamint eddig k\"uld\"ottem, tsak
k\'artyah\'az a toronyhoz k\'epest\dots''
\end{quote}
Or, translated from the 19th century Hungarian:
\begin{quote}
``I deduced things so marvellous that I was enchanted myself, and it would
be an eternal loss to let them pass; Dear Father, once you see them, you
will recognise their greatness yourself; now I cannot tell you more, only
this: out of the void I created a new, a different world; all that I sent
you before is like a house of cards to a strong tower\dots''
 \end{quote}

The discovery of non-Euclidean hyperbolic geometry\index{non-Euclidean
geometry}\index{hyperbolic!geometry} was indeed a landmark in modern
scientific thinking, as revolutionary and as far reaching in its
implications as the Copernican model of the solar system or Darwin's theory
of evolution. For an account of the very interesting history, see Greenberg
\cite{Gg}. The early models of hyperbolic geometry were abstract; simple
coordinate models, such as that used in Chapter~\ref{ch!hyp} of this course,
were developed later in the second half of the 19th and the early 20th
century. Hyperbolic geo\-metry soon found many applications in different
areas of mathematics and science; in particular, the notion of curved space
in differential geometry plays a foundational role in Einstein's general
relativity (1916).

Spherical geometry seems to have been excluded from consideration in
descriptive or axiomatic geometry from the time of Euclid for two reasons.

 \begin{enumerate}\renewcommand{\labelenumi}{(\alph{enumi})}
 \item More obviously, any two lines meet in two points (a pair of antipodal
 points); this is not a very serious defect, because you can pass to the
 geometry of $S^2/\{\pm1\}=\PP^2_\R$, in which every pair of lines meets in
 just one point.
 \item Its lines do not satisfy the {\em order} condition implicit in
 Euclid: given three points $P$, $Q$, $R$ on a spherical line (great
 circle), it is impossible to say which of the three is ``between'' the
 other two. Equivalently, a point $P$ of a spherical line (great circle)
 does not divide it into disconnected sets. That is, given a line $L$ and a
 point $P$ not on it, {\em every} line $M$ through $P$ meets $L$ {\em both}
 over there to the left {\em and} over there to the right (see
 Figure~\ref{sph_parallel_axiom}). In spherical
 geometry\index{spherical!geometry} these are antipodal points; in the
 geometry of $S^2/\{\pm1\}=\PP^2_\R$, the same point.
 \end{enumerate}

 \begin{figure}[ht]
 \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}c}
 \centering \input{final_sph_para_post.pstex_t}
 \caption{The ``parallel postulate'' in spherical geometry}
 \label{sph_parallel_axiom}
 \end{figure}

Euclid's axioms did not discuss the separation properties of points on a
line: it was supposed to be understood what it meant for $A$ to be between
$P$ and $Q$ on the line segment $PQ$. (Compare the discussion
of~\ref{sec!separation}; separation is a topological statement about the
geometry.) Thus it is not surprising that spherical geometry was overlooked;
however, this is a fair indication that Euclid's claim to rigour in a modern
sense was never really watertight.

Nevertheless, spherical geometry has been around in an ``applied'' form for
centuries. Spherical trigonometry\index{spherical!trig} has been studied
very thoroughly in the context of navigation since the time of Newton, to
aid ships sail around the oceans of the world. Because of winds and currents
though, the lines of spherical geometry, great circles, are not always the
fastest way to travel. These days, great circles are the routes taken for
preference by airlines, except when no-fly zones intervene.

\subsection{Coordinates versus axioms}

Descartes' invention of coordinate geometry is another key ingredient in
modern science. It is scarcely an accident that calculus was discovered by
Leibnitz and Newton (independently, alphabetical order) in the fifty years
following the dissemination of Descartes' ideas. Interactions between the
axiomatic and the coordinate-based points of view go in both ways:
coordinate geometry gives models of axiomatic geometries,\index{axiomatic
projective geometry} and conversely, axiomatic geo\-metries allow the
introduction of number systems and coordinates. There are several excellent
books giving systematic treatments of these very interesting issues; the
classic~\cite{Hil} is warmly recommended.

As in art or music or politics, attitudes and fashions in mathematics vary
quite sharply from one generation to the next. In the second half of the
19th century, up to the time of Hilbert and Poincar\'e, geometry was
without doubt at the centre of mathematics and of large areas of theoretical
physics. This position was overturned with the rise of abstract algebra,
topology and set theoretic foundations of mathematics around the 1920s. The
blame for this lies in part with the geometers themselves, who developed a
sloppy attitude to correct statements and proofs of theorems. One example is
the type of argument that involved ``sufficiently general position'', which
might in favourable cases have a precise meaning within an epsilon
neighbourhood of the author. In England, there was a brilliant school of
geometers between the wars in Cambridge, which seems to have been broken up
when the participants were drafted into code breaking or aeronautics during
the second world war. When the senior author was an undergraduate at
Cambridge (late 1960s), geo\-metry in the sense of this course was
universally considered a terribly dull fuddy-duddy subject. The position has
been entirely turned around in the last 30 years, and at present geometry in
its various manifestations again claims centre stage in mathematics and
theoretical physics.

\section{Group theory}\label{sec!gp_thy}

\subsection{Abstract groups versus transformation groups}\label{ssec!gp_acts}

According to the abstract definition (which is comparatively recent), an
abstract group is a set with a composition law satisfying a couple of well
known axioms. However, from the beginnings of the subject in the 19th
century, the groups studied were always thought of as symmetry groups,
that is, as transformation groups preserving some structure or other. For
example, Ruffini, Abel and Galois considered permutations of the roots of a
polynomial equations, and the subgroup of permutations that preserve the
rules of arithmetic. From the mid-19th century, many other groups arose as
geometric symmetries: finite groups such as the symmetries of the regular
polyhedra, infinite but discrete groups in the study of crystallography,
that contain translations by a lattice as a subgroup, and Lie groups such
as the Euclidean group. The idea that a group can be treated as an abstract
composition law {\em without reference to the nature of the operators that
make it up} was first introduced by Cayley in 1854, but its significance
was not recognised until much later.

Let $G$ be a group and $\Si$ a set; I say that $\Si$ is a {\em $G$-set} or
that {\em $G$ acts on $\Si$}, if a group homomorphism
 \[
 \fie\colon G\to\Trans\Si
 \]
is given from $G$ to the group of transformations of $\Si$ (see~\ref{transf
as group}). That is, each $g\in G$ corresponds to a transformation
(bijective map) $\fie_g\colon\Si\to\Si$, in such a way that the abstract
composition law in $G$ corresponds to composition of transformations of
$\Si$. In other words, $G$ is trying to fulfil its destiny as a
transformation group of $\Si$, as discussed in Chapter~\ref{ch!gp}. One
usually writes simply $\fie_g(x)=gx$ or $g(x)$ for the action of $g\in G$
on $x\in\Si$.

The requirement that the map $\fie$ is a homomorphism is written
$(gh)x=g(hx)$. This looks like an associative law, but the meaning is just
that the abstract product in $G$ corresponds to composition of maps
$\Si\to\Si$; compare the discussion in \ref{sec!assoc}. Evaluating $g\in G$
on $x\in\Si$ provides a map $\Phi\colon G\times\Si\to\Si$ given by
$\Phi(g,x)=\fie_g(x)$; I leave it to you to express the condition
$(gh)x=g(hx)$ in these terms.

\subsection{Homogeneous and principal homogeneous spaces}\label{ssec!torsor}

\begin{defn} Let $\Si$ be a $G$-set. I say that $G$ acts {\em transitively}
on $\Si$ if the action takes any point of $\Si$ to any other. In this case
$\Si$ is a {\em homo\-geneous space} under $G$. \end{defn}

This idea has already appeared many times: the geometries in the earlier
chapters of the book were homogeneous under appropriate groups. For example,
the Euclidean group acts transitively on $\E^n$: any point of $\E^n$ goes to
the origin under a suitable Euclidean motion. The affine group $\Aff(n)$
acts transitively on {\em pairs} of distinct points of $\aff^n$; as
discussed at several points of the book, this is closely related to the fact
that affine geometry does not have an invariant distance function.

If $\Si$ is a $G$-set and $x\in\Si$, the {\em stabiliser subgroup} of $x$ is
the set of elements of $G$ that fix $x$, that is
 \[
 \Stab_G(x)=\bigl\{h\in G\bigm| h(x)=x\bigr\}.
 \]
For example, the stabiliser subgroup of the origin $0\in\E^n$ in $\Eucl(n)$
is the group $\O(n)$ of orthogonal matrixes.

If $G$ acts transitively on $\Si$, the map $e_x\colon G\to\Si$ defined by
$g\mapsto gx$ is surjective. Moreover elements $g_1,g_2\in G$ map to the
same point of $\Si$ if and only if $g_2=g_1h$ for some $h\in\Stab_G(x)$;
thus $e_x$ induces a bijection $G/\Stab_G(x)\isoto\Si$. (Here $G/H$ stands
for the quotient of $G$ by the equivalence relation $g\sim gh$ for $h\in H$,
or the set of left cosets.)

\begin{defn} A homogeneous space $\Si$ under $G$ is a {\em principal
homo\-geneous space under} $G$ or a {\em $G$-torsor} if the stabiliser
$\Stab_G(x)$ is trivial for every $x\in\Si$. Since the stabiliser of $x$ and
$gx$ are conjugate (by the same argument as in
Ex.~\ref{ch!gp}.\ref{ex!stab}), it is enough to verify that $\Stab_G(x)$ is
trivial for any one $x\in\Si$. \end{defn}

For example, affine space $\aff^n$ is a homogeneous space under $\Aff(n)$,
but is a torsor under the translation subgroup $\R^n\subset\Aff(n)$.

According to the previous discussion, if $\Si$ is a $G$-torsor, then
$e_x\colon G\to\Si$ is a bijection from $G$ to $\Si$, and I could use this
to identify $G$ and $\Si$. However, different elements of $\Si$ give
different bijections: the set $\Si$ has no distinguished identity element.

\begin{exa}
Let $\Si$ consist of the vertexes of a regular $n$-gon in the plane~$\E^2$,
$G\subset\Eucl(2)$ the group of symmetries of $\Si$ (the dihedral group
$D_{2n}$, see Exercise~\ref{ch!gp}.\ref{ex!dihed}), 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 $\Si$ is transitive, since the
polygon is regular. Thus $\Si$ is a homogeneous space under $G$. The
stabiliser $\Stab_G(P)$ of a vertex $P\in\Si$ is of order two, consisting of
the identity and the reflection in the axis through $P$. The subgroup $H$
acts transitively and without stabilisers (since it does not contain
reflections). Thus $\Si$ is an $H$-torsor: there are as many vertexes as
rotations, but no vertex is distinguished over the others.
\end{exa}

\subsection{The Erlangen program revisited}\label{subsec!Erlangen_re}

Recall Klein's Erlangen program of Section~\ref{sec!erlan}, the slogan that
geometry is the study of properties invariant under a transformation group
$G$. The introduction to Chapter~\ref{ch!eucl} discussed the basic geometric
and philosophic principles: space should be
 \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})}
 \item homogeneous (the same viewed from every point), and
 \item isotropic (the same in every direction).
 \end{enumerate}
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. Helmholtz' 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
{\em 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
\begin{itemize} \renewcommand{\labelenumi}{(\arabic{enumi})}

\item Euclidean space $\E^n$ is a homogeneous space under the Euclidean
group $\Eucl(n)$. The stabiliser of a point $P\in \E^n$ is isomorphic to the
group $\O(n)$, the group of rotations and reflections fixing $P$. (It is the
group of orthogonal matrixes in the $\R^n$ that is the orthogonal complement
of $OP$.) By Theorem~\ref{Eframe}, the set of Euclidean frames forms a
torsor under $\Eucl(n)$.

\item The sphere $S^n$ is a homogeneous space under the group $\O(n+1)$ of
spherical motions (Theorem~\ref{s2d} for $n=2$; the general case is
identical). For $P\in S^n$, the stabiliser group is isomorphic to the group
$\O(n)$.

\item Hyperbolic space $\sH^n$ is homogeneous under the Lorentz group
$\O^+(n,1)$. The stabiliser of a point $P$ is again isomorphic to the group
$\O(n)$.

\item Projective space $\PP^n$ is homogeneous under the projective linear
group $\PGL(n+1)$. The stabiliser of a point $P$ is isomorphic to $\PGL(n)$.
By Theorem~\ref{PGL}, the set of projective frames of reference forms a
$\PGL(n+1)$-torsor.

\end{itemize}

\subsection{Affine space as a torsor}\label{subsec!affdef}

The notion of torsor formalises the ad hoc definition of affine space I gave
in Chapter~\ref{ch!aff}. Let $V$ be a vector space; an affine space
$\aff(V)$ is just a torsor under $V$. In other words, $\aff(V)$ is a set
with an action of $V$ (``by translation''), and this action is simply
transitive: for $P,Q\in V$ there is a unique vector $\bx\in V$ such that
$Q=P+\bx$.

Looking back to~\ref{sd prod}, I can say all this slightly differently: the
trans\-formation groups in Euclidean and affine geometry are semidirect
products. For example, the Euclidean group
 \[
 \Eucl(n) = \O(n)\ltimes \R^n
 \]
is the semidirect product of the normal subgroup of translations and the
group of rotations. From the analysis of~\ref{sd prod}, it follows that the
subgroup $\O(n)$ is not normal. The conjugation construction (see
\ref{cjgcy}) allows me to {\em define} Euclidean space to be the space of
all conjugates of a fixed copy of $\O(n)\subset\Eucl(n)$, and notions of
Euclidean geometry to be all notions that can be defined on this space
invariantly under the group $\Eucl(n)$. This is of course the Erlangen
program repeated once again.

I can say the same words starting from the group of affine transformations
$\Aff(V)$ (see~\ref{sec!afflinA}). This contains copies of $\GL(V)$, the
group of invertible linear maps of $V$, as affine transformations fixing a
point, and these subgroups are once again non-normal.

% I can thus {\em define} affine space $\aff^n$ to be the set of all
% conjugates of a fixed copy of $\GL(V)\subset\Aff(V)$, giving finally a solid
% definition of affine space in place of all the wobble
% in~\ref{sec!affbasics}.

From the group theory it follows then that the group of translations $V$
acts transitively with trivial stabiliser on $\aff(V)$, thus $\aff(V)$ is a
$V$-torsor (a principal homogeneous space under the group of translations).
In other words, we have an action $\fie_\bv\colon P\mapsto P+\bv$ of the
additive group of $V$ defined on points of affine space. For $P\in\aff(V)$,
we get a bijection $e_P\colon V\to \aff(V)$ mapping $\bv\in V$ to
$P+\bv$; two such identifications differ by an element of $V$ acting by
translation. The bijections $e_P$ are different coordinate systems on affine
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 $P,Q\in\aff(V)$
determine a vector $e_P(Q)=\vec{PQ}\in V$ (cf.~Figure~\ref{points of An}).

The point here is that in 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 $\O(n)$ is given, $\E^n$ is the
homogeneous space $\Eucl(n)/\O(n)$, where $\O(n)=\Stab(x)$; alternative,
$\E^n$ is the set of subgroups conjugate to $\O(n)$.

 \section{The famous trichotomy}
 \markright{\hfill \thesection\ Trichotomy\quad}
 \label{sec!trichotomy}

The metric geometries of this course come in a triad: spherical, Euclidean
and hyperbolic. The geometries correspond to the local notion of positive,
zero or negative curvature. You can determine which geometry you are in
locally by measuring the perimeter of a circle of radius $R$, which comes
out to be $2\pi\sin R$, $2\pi R$ and $2\pi\sinh R$, in the three cases (see
Figure~\ref{fig!3tomy} and refer to Exs.~\ref{S2circle}
and~\ref{ex!circetal}).
 \begin{figure}[ht] \label{fig!3tomy}
 cap, flat plane and Pringle's chip
 \caption{The cap, flat plane and Pringle's chip: the perimeter of a circle
or area of a disc of radius $R$ determines the local curvature of space.}
 \end{figure}
Globally, as discussed above, 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 exponentially.

Topologically, the Euclidean plane $\E^2$, the sphere $S^2$ and hyperbolic
space $\sH^2$ are all simply connected (cf.~\ref{wind}; for $\sH^2$, use the
homeomorphic model $\sH$ of~\ref{ch!hyp}.\ref{ex!defH}--\ref{ex!hypgraphics}
if you wish). As well as these simply connected geometries however, we can
also consider compact ones; for simplicity we only discuss the oriented
surfaces 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 identifies points which are related to each other by
translation by vectors in a fixed parallelogram 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 more than one hole can be derived from the
hyperbolic plane (Figure~\ref{fig!surf}).
\begin{figure}[ht]
 \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}}
 \centering \input{final_surf.pstex_t}
 \caption{The famous trichotomy}
 \label{fig!surf}
 \end{figure}
The number of holes in a compact surface is called its {\em genus}; so in
terms of the genus, our trichotomy becomes $g=0$, $g=1$ or $g>1$. To return
to the basic trichotomy of positive, zero or negative curvature, we can take
the Euler number $\chi=2-2g$ of the surface, which is simply the quantity
``$\mathrm{faces}-\mathrm{edges}+\mathrm{vertexes}$'' in Euler's\index{Euler
characteristic} formula for a triangulated surface. Then $\chi=2$ for a
sphere, as every baby knows; also $\chi=0$ for a torus and $\chi<0$ for the
geometric surfaces with more than one hole. As a fun exercise, try to
triangulate a surface with two holes (compare Ex.~\ref{top
chap}.\ref{ex!genus2}), and check Euler's formula!

The classification of three dimensional geometries that extend our
curvature trichotomy in two dimensions goes by the name of Thurston's
Geo\-metrisation Conjecture (late 1970s). This includes as a humble first
case the Poincar\'e Conjecture characterising the 3-sphere; it seems this
may well become the first to be solved of the Clay Mathematical Institute's
million-dollar Millennium Prize Problems. In a different direction, my own
subject of classification of varieties in algebraic geometry studies
geo\-metric shapes defined in space by several polynomial relations; the
curvature trichotomy re-appears there in an algebraic form.

Much was written up to the turn of the 20th century on the subject of
whether our own three dimensional universe is Euclidean, spherical or
hyperbolic; Poincar\'e's extended essay {\em La science et l'hypoth\`ese}
(1902) points out that the question itself begs a number of conventions, for
example on how the objects of geometry (straight lines, distance) are
realised as physical objects (light rays, observations of astronomy). Maybe
the answer to the question depends on our choice of conventions. The
universe has grown in size and complexity since Poincar\'e's day. According
to special relativity (1905), it does not make sense to consider space as a
separate entity from space-time. General relativity (1906) says that
space-times is not flat or even of constant curvature, but is curved by the
presence of matter, thus resolving the instantaneous action-at-a-distance that
was a philosophical contradiction implicit in Newton's theory of
gravitation. The existence of black holes seems to be acknowledged by the
majority of astrophysicists and cosmologists; they can presumably only be
represented in geometry as singularities localised around isolated points.
One could believe that the effects of general relativity can be envisaged as
merely minor localised disturbances, and that space in the large is
nevertheless Euclidean; this is possibly the view held by many practising
cosmologists (I have not carried out a scientific survey). However, the same
population will cheerfully admit that something like 80\% of the mass of the
universe seems not to be accounted for by current theories (``black matter''
and ``black energy''). Just a little overall curvature or cosmological
constant could go a long way (compare Ex.~\ref{ex!pringle}).



% The future of the universe -- big crunch

 \section{Geometry in physics}
 \markright{\hfill \thesection\ Geometry and physics\quad}
 \label{sec!geom&phys}
Some of the most substantial applications of geometric ideas
come from physics. Recall the grandiose aim expressed in my first sentence:
 \begin{quote}
 ``Geometry attempts to describe and understand space around us and all that
 is in it.''
 \end{quote}
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-simplified; at the end I give references to the literature for further
study.

\subsection{The Galilean group and Newtonian dynamics}
\label{sec!galilei}

The dynamics of Galileo and Newton takes Euclidean three space $\E^3$ as the
fundamental model of physical space, and time $t$ as a universal parameter
with a preferred directionality. Thus space-time is modelled by
$\E^3\times\R$, with coordinates $(\bx, t)$. Spatial lengths are measured
with respect to the Euclidean metric~\ref{Rn}, and involve only the
$\bx$-coordinate; events also have a time separation $t_2-t_1$ (no absolute
value is taken here). Valid coordinate systems describing Newtonian
mechanics 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
 \[
 (\bx, t) \mapsto (A\bx+ \bg t + \bb, t+s),
 \]
where $A\in\O(3)$ is a $3\times3$ orthogonal matrix, $\bg$ and $\bb$ are
$3\times 1$ column vectors, and $s\in\R$ is a scalar. Such transformations
collectively form the {\em Galilean group} $\Gal(3,1)$ of classical
$(3+1)$-dimensional space-time $\E^3\times \R$. 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
 \[
 (\bx, t) \mapsto (A\bx+\bb, t),
 \]
with $\bg=0$ and $s=0$. Transformations with nonzero $\bg$ correspond to a
change to a new reference frame in uniform movement of speed $\bg$ with
respect to the old one; such group elements are usually called {\em Galilean
boosts}. Elements of ${\rm Gal(3,1)}$ with $s\ne 0$ correspond to moving the
origin of time; Newtonian physics has no fixed Creation or Big Bang. It is
however not possible to stretch or reverse time, however you might wish to
during the exam.

The shape of the Galilean group determines Newton's equation of motion,
in the form familiar to you from a first Mechanics course. For a single
particle with mass~$m$ and position vector $\bx(t)$ at time~$t$, with no
external forces acting, the equation simply says
 \[
 m\frac{\dd^2\bx(t)}{\dd t^2}=0.
 \]
Note that this equation is indeed invariant under the Galilean group.

Emmy Noether'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 $\bx_i(t)$,
Table~\ref{tab!conserve} describes the conserved quantities of Newtonian
mechanics.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.1}
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\thetable}{\arabic{chapter}.\arabic{section}}\label{tab!conserve}
 \begin{center}
\begin{tabular}{|l|c|l|}
 \hline
Symmetry & $\begin{array}{c}\hbox{Conserved}\\ \hbox{quantity}\end{array}$ & Name \\
 \hline
 \hline
$\begin{array}{l}\mbox{spatial translation}\\(\bx, t)\mapsto(\bx+\bb,
t)\end{array}$ & $\displaystyle\sum_i m_i \displaystyle\frac{\dd\bx_i}{\dd
t}$ & momentum \\[12pt]
 \hline

$\begin{array}{l}\mbox{spatial rotation}\\(\bx, t)\mapsto(A\bx,
t)\end{array}$ & $\displaystyle\sum_i m_i \bx_i \times
\displaystyle\frac{\dd\bx_i}{\dd t}$ & angular momentum \\[12pt]
 \hline

$\begin{array}{l}\mbox{Galilean boost}\\(\bx, t)\mapsto(\bx+\bg t,
t)\end{array}$ & $\displaystyle - \bp t + \sum_i m_i\bx_i$ &
$\begin{array}{l}\mbox{center of mass (where $\bp$ is} \\ \mbox{the total
momentum)} \end{array}$ \\[12pt]
 \hline

$\begin{array}{l}\mbox{time translation}\\(\bx, t)\mapsto(\bx,
t+s)\end{array}$ & $\displaystyle\sum_i \displaystyle\half
m_i\Big|\frac{\dd\bx_i}{\dd t}\Big|^2$ & energy \\[12pt]
 \hline
\end{tabular}
 \end{center}
\caption{Symmetries and conservation laws}
\end{table}

\subsection{The Poincar\'e group and special relativity}

Newtonian dynamics functioned well as a description of space-time up until
the late 19th century. At that time however, two new developments shattered
its foundations. The first nail in its coffin was the famous
Michelson--Morley experiment (1887), which refuted the best current
explanation of the properties of light within Newtonian theory in terms of
the ``theory of ether''. The simplest interpretation of their result was
that the speed of light was independent of the speed of the observer, in
stark contradiction with the Galilean group, which obviously cannot
accommodate such behaviour. A second (closely related) fact involves
Maxwell's equations of electromagnetism, which are not invariant under the
Galilean group.

After an exciting decade of developments, best summarised elsewhere,
Einstein's 1905 foundational paper spelled out a new theory, {\em special
relativity}, based on a different set of principles. Four dimensional
space-time is henceforth to be modelled on $\R^{1,3}$, which is shorthand
for a space with coordinates $\bx = (t,x_1,x_2,x_3)$ and {\em Lorentz
metric}
 \[
 \dd s^2=c^2\dd t^2-\dd x_1^2-\dd x_2^2-\dd x_3^2;
 \]
or, if you are not familiar with infinitesimal notation,
you can write the {\em Lorentz distance}
of vectors $\bx=(t,x_i), \by=(s, y_i)\in\R^{1,3}$ as
 \[
 d( \bx, \by) = c^2(t-s)^2- \sum_i(x_i-y_i)^2.
 \]
Here the constant $c$, with (classical) dimensions length/time, is the {\em
speed of light}, postulated to be universal in all 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 $\R^{1,3}$ (and the ``positive
light-cone'') is preserved; this is {\em Einstein's relativity principle}.
The group of such transformations is the {\em Poincar\'e group}\footnote{The
naming of concepts during these exciting years was rather haphazard, often
respecting accident and scientific standing more than historical accuracy.
In particular, the so-called Lorentz metric appears to have been proposed
first (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. Poincar\'e came very close to inventing
special relativity in the years 1900--1904, showing in particular that
Lorentz transformations form a group; hence in the case of the Poincar\'e
group, the name is accurate.} $\Poin(1,3)$ consisting of maps
 \[
 \bx\mapsto A\bx + \bb,
 \]
where $A\in \O^+(1,3)$ is a Lorentz matrix (preserving the positive cone),
and $\bb\in\R^{1,3}$. This group can be studied in complete analogy with the
treatment of~\ref{sd prod}: it is the semidirect product
 \[
 \Poin(1,3)\cong\O^+(1,3)\rtimes\R^{1,3}
 \]
of a normal subgroup, the four dimensional Lorentz group $\O^+(1,3)$, and
the group $\R^{1,3}$ of space-time translations. Also, for fixed values of
the time variable~$t$, the metric reduces to (the negative of) the Euclidean
metric on a copy of $\R^3$. Hence $\Poin(1,3)$ contains a subgroup
$\Eucl(3)$ of Euclidean transformations. However, since the Poincar\'e group
mixes $t$ and $x$ coordinates, this splitting of space-time into ``time''
and ``space'' is not canonical, but depends on the choice of coordinate
frame (observer).

Hyperbolic geometry is contained in the Lorentz space $\R^{1,n}$ of special
relativity as the space-like hypersurface
 \[
 q_L(t,x_i)=1, \quad\hbox{with $t>0$.}
 \]
The distinction of time-like and space-like vectors in the Lorentz model of
hyperbolic geometry derive their name exactly from this physical interpretation.

\subsection{Wigner's classification: elementary particles}
\label{sec!wigner}

As discussed above, the Poincar\'e group $\Poin(1,3)$ contains the Euclidean
group $\Eucl(3)$, hence also the Euclidean rotation group $\SO(3)$. As you
recall from~\ref{sec!topson}--\ref{sec!su2}, the latter group has a double
cover $\SU(2)\to \SO(3)$, that is, a 2-to-1 surjective group homomorphism with
kernel $\pm1$. It turns out that this double cover extends to a double cover
 \[
 \wPoin(1,3)\to \Poin(1,3)
 \]
of the Poincar\'e group, which can be constructed using the group
$\SL(2,\C)$ of $2\times2$ complex matrixes of determinant~1 (which obviously
contains the group $\SU(2)$ covering $\SO(3)$).

One of the first spectacular uses of group theory in theoretical physics was
Wigner's insight of the 1940s, which relates ``symmetries of space-time'' to
``things in it'' (particles), and can be summarised as follows
(see Sternberg [17] for the physical intuition and more details):
\begin{enumerate}\renewcommand{\labelenumi}{(\arabic{enumi})}
\item An ``elementary particle'' of nature is an (finite dimensional,
irreducible, unitary) representation of the symmetry group of space-time,
satisfying certain ``physical restrictions''.
\item The symmetry group of space-time is the Poin\-car\'e group, or more
precisely its universal cover $\wPoin(1,3)$.
\item The classification of the relevant representations of the Poincar\'e
group thus leads to a classification of all elementary particles.
\end{enumerate}
Recall from~\ref{sec!lie} 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 ``physically relevant'' representations
of $\wPoin(1,3)$ are classified by
\begin{itemize}
\item a continuous nonnegative parameter $m\ge 0$, called the
{\em rest mass} of the particle, and
\item a half-integer $s$, called {\em particle spin}, that is allowed to
takes nonnegative values $0,\half,1,\dots$ for particles of mass $m>0$, and
all values $0,\pm\half,\pm 1,\dots$ for those with $m=0$.
\end{itemize}
Integral spin particles correspond to representations for which the kernel
$\pm1=\ker\wPoin(1,3)\to \Poin(1,3)$ 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 {\em
photons}, which are massless (that is, $m=0$) and have integral spin $s=1$,
and electrons with $s=\half$ and a certain positive value of~$m$. (The
phenomenon of spin~$\half$ particles was the main point of the discussion of
\ref{sec!spin}.) The group $\wPoin(1,3)$ has additional ``nonphysical''
representations with $m^2<0$; these are called {\em tachyons} (mythical
particles travelling faster than the speed of light), and are relegated to
the world of science fiction in most current theories (but not all).

\subsection{The Standard Model and beyond}
\label{sec!stdmodel}

The importance of Wigner's insight in the development of modern physics can
hardly be overstated: in a sense, it concludes another 2000 plus year old
story, the search for the ultimate building blocks of the physical universe.
Of Wigner's program, (1) and (3) have stood as cornerstones of most theories
of particle physics proposed in the last 50 years. Only (2), the specific
choice of the symmetry group, has changed during the course of subsequent
developments.

One thing that was clear already at the outset is that Wigner's original
discussion does not incorporate the electromagnetic interactions of
elementary particles. This however only requires a minor modification,
taking into account an additional {\em internal} symmetry group $\U(1)$.
This group is no longer a geometric symmetry of space-time, but rather a
symmetry of the whole theory of electromagnetism in space-time, used to
encode additional data. Representations of the combined group
$\wPoin\times\U(1)$ are now parametrised by a triple of numbers $(m,q,s)$,
with the additional quantum number $q$, the {\em electric charge}, taking
integer values.

As the particle accelerators of the 1950s and 1960s grew capable of
producing faster and faster particles and then slamming them into one
another with more and more energy, 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 {\em 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 confirmed in subsequent accelerator
experiments. There is really no space here to elaborate on this development;
I recommend~\cite{St} as a good source. Let me only say that currently, the
most popular, the {\em Standard Model}, is based on the Poincar\'e group
augmented by the internal symmetry group $\U(1)\times\SU(2)\times\SU(3)$;
roughly, the three factors are responsible for the electromagnetic, weak and
strong forces (this is of course a gross over-simplification).

Embedding the internal symmetry group $\U(1)\times\SU(2)\times\SU(3)$ into
an even larger group, mixing all three forces (electromagnetic, weak and
strong) completely, come under the name {\em Grand Unification Theory}
(GUT), a sometime favourite pastime of ``armchair physics''. Popular GUT
groups include the special unitary group $\SU(5)$, the group $\SO(10)$, and
even more exotic constructs such as the 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.

\subsection{Other connections}

The connections between geometry and physics extend beyond the relationship
between space-time symmetries and particles. The two crowning achievements
of early 20th century physics, quantum theory and general relativity, are
inextricably linked to geometric ideas in several ways. The influence 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 Einstein's insight that gravity, instead of acting
as a classical ``force'', is better described as encoded into the local
curved structure of space itself. Quantum mechanics, invented by
Schr\"odinger and Heisenberg in the 1920s, was axiomatised by Dirac and von
Neumann, building on the Hilbert incidence axioms for projective geometry
(see~\ref{Axiomatic projective geometry}).\index{axiomatic projective
geometry} 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.

 \section{Further directions}\label{sec!titles}
 \markright{\hfill \thesection\ Essay titles\quad}

\dots and finally, a bulled-point list of other exciting directions,
connections and developments which I omit for lack of space, expertise or
both. Many of these would make excellent topics for undergraduate essays.

 \begin{enumerate}\renewcommand{\labelenumi}{(\arabic{enumi})}

 \item The classification of locally Euclidean geometries in the style of
Nikulin and Shafarevich~\cite{NSh}.

 \item History of navigation, trigonometry and spherical geometry.
\index{spherical!geometry}

 \item \label{entry!maps}
 Spherical geometry, projections and cartography (map making).

 \item $\SO(3)$ and Euler angles, mechanics in moving frames, Coriolis forces.

 \item Plane and spherical geometry and plate tectonics: why South America
 and West Africa fit together like pieces of a spherical jigsaw puzzle.
 Classification of fault types and Euler's theorem. See for example [C.M.R.
 Fowler, {\em The solid earth}, CUP 1990, Chapter~2].
\index{spherical!geometry}

 \item Symmetry groups in geometry. This is a vast subject, with relations
to crystallography~\cite{Cx, NSh}, Escher's art, regular polyhedra
and Penrose tilings.

 \item Subgroups of the symmetric group in puzzles and toys. Examples include
the perfect shuffle groups and moves of the Rubik cube; start for example
in Chapter 19 of P.M. Neumann, G.A. Stoy, E.C. Thompson, {\em Groups
and geometry}, OUP, 1994.

 \item Axiomatic projective geometry,\index{axiomatic projective
geometry} leading to von Neumann, foundations of quantum theory,
$\C^*$ algebras and so on.

 \item Geometry and mechanics: Newton's equations, planetary
motion and conics.

 \end{enumerate}

