%quotient spaces with a topology
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\newtheorem{fact}[theorem]{Fact}
\newtheorem{GaussBonnet}[theorem]{The Gauss--Bonnet Theorem}
\newtheorem{ThurstonConj}[theorem]{The Thurston Geometrization Conjecture}
\newtheorem{PerelmanProof}[theorem]{Perelman's Proof}
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\title[Perelman Proves Poincar\'e]{Perelman Proves Poincar\'e\\
\ \\
\ \\
\includegraphics{MCEHandH.eps}}
\author{Jonathan A.~Poritz}
%\address{\tt www.poritz.net/jonathan}
%\email{jonathan.poritz@gmail.com}
\begin{abstract}\noindent
In 1904, {\bf Henri Poincar\'e} conjectured:\hfil\break
\indent\indent \textsl{\Large Every compact, connected, simply-connected $3$-manifold
is the $3$-sphere.}
\vskip2mm
This conjecture has been near the center of a maelstrom of activity in
topology, geometry, analysis, and many allied and sub-disciplines for
a hundred years. Recently, Grisha Perelman announced a proof of the
conjecture -- in fact of the stronger {\bf Geometrization Conjecture} of
{\bf William Thurston} (from 1982), which states:\hfil\break
\indent\indent \textsl{\Large Every prime, orientable and compact
$3$-manifold can be cut along tori into pieces}\hfil\break
\indent\indent \textsl{\Large each of which has a finite volume
geometric structure from the Eightfold Path.}
\vskip2mm
[See below for more (and more meaningful) details).] In this paper I
collect many of the definitions and a few of the theorems which an
interested mathematical tourist will need to understand the outlines,
at the least, of the progress made in this area. The goal here is to
have fairly precise and accurate definitions (and pointers to some
literature) which would help someone who became deeply interested in
this area to begin the lifetime of study that would inevitably ensue.
\end{abstract}
\date{\thisdaymonthyear\hfil
{\tt www.poritz.net/jonathan}\hfil
{\tt jonathan.poritz@gmail.com}}
\maketitle
%\setcounter{page}{1}
\setcounter{section}{-2}
\section{A Dream (A Nightmare?)}\label{sec:dream}
You are in a dark room in a shabby hotel. You hear someone come into
the room. They hand you a strange object and demand, in a heavy French
accent, to know what it is. You feel it carefully, trying to understand.
You can certainly feel that it is three-dimensional [\emph{it's a
$3$-manifold}]. You pinch one corner and hold it up, no pieces fall off
[\emph{it's connected}]. You cannot feel any sharp edges, nor does it trail
out the door or window [\emph{it's compact}]. You think of hanging
it from a hook in the ceiling so as to get a better gander at it with both
hands, but it is very squishy and flexible so you cannot tie a string merely
around a protrusion [\emph{we are interested in identification up to
homeomorphism or diffeomorphism (or PL-diffeomorphism), which are rather
flexible notions, not at all rigid like an isometry would have to be}], nor
can you find a part of the object through which to thread your string -- any
loop of string you put on the object shrinks to nothing and the object falls
to the floor [\emph{it's simply-connected}].
The walls of the hotel are atrociously thin. Next door, you hear the
hotel guest (whose name, you noticed in the ledger, is Stephen Smale)
whistling ``The Girl from Ipanema'' and laughing about how easy it is
to untangle highly twisted disks when you can untwist in many dimensions.
He seems to be saying he settled with the Frenchman. [\emph{Smale, and,
shortly after, Stallings and Wallace, proved the Poincar\'e conjecture
in dimensions $5$ and higher.}]
On the other side, you hear very strange sounds, as though someone were
assembling a chopped-liver swan. Another guest (Michael Freedman -- you
also noticed \emph{his} name) seems to be working very hard, and to great
effect, on very squishy objects. [\emph{Freedman proved the $4$-dimensional
topological Poincar\'e Conjecture, as part of a complete classification of
the homeomorphism types of $4$-manifolds by their intersection forms and
Kirby-Siebenmann invariants.}] A British accented voice in that room
shouts ``What about this!?!'', whereafter there is great commotion and
then relative silence. [\emph{Simon Donaldson proved the existence of a
$\RR^4_\textrm{fake}$ which is homeomorphic, but not diffeomorphic, to the
usual $\RR^4$; Gompff and others then found many more such strange examples,
but their complete classification is still mysterious.}]
Someone new comes into your room, smelling of surfboard wax and talking
about ``going with the flow''. He seems to have a good idea what your
mystery object is, but he gets nervous when you tell him you weren't very
positive about the object from the beginning. [\emph{Richard Hamilton
used the Ricci flow to prove important results in dimension three, but his
strongest results were in the case of manifolds of everywhere positive Ricci
curvature.}] He retreats to the corner of your room, muttering about cigars
and pencil-neck geeks. [\emph{Hamilton's progress was stopped by worries
about a stable (and useless) 'cigar soliton' or thin necks appearing in
finite time in the Ricci flow.}]
A noisy crowd fills the hallway for a while, you hear them shouting for
someone named ``Wild Bill''. This Bill may be a Buddhist (or just drunk),
because they are all talking loudly about an ``eightfold way'', and he
is boasting that the Frenchman is completely taken care of by one of these
eight ways. [\emph{William Thurston generalized the Poincar\'e conjecture
to a proposed classification of all $3$-manifolds -- called the
``Geometrization Conjecture'' -- consisting of a decomposition of these
manifolds into canonical simple pieces, each of which must then admit one
of eight particular geometries; from this would follow Poincar\'e without
trouble. Thurston advanced this theory on many fronts, but seemed to have
no approach for the complete proof.}]
After what seems like a hundred years, a very quiet Russian enters your
room. He has Bill in tow, and he reassures the fearful surfer that everything
is OK. Standing (gently) on the surfer's shoulders, he repairs the light
fixture in your room, and you now have enough illumination [\emph{thanks
to people like Morgan, Tian, Kleiner and Lott}] to see that your mystery
does indeed flow, that it has no cigars and the necks it does develop are
not to be feared -- in fact, the necks point right at Bill's eightfold
way. That damn Frenchman is gone but your mystery object is just a $3$-sphere.
[\emph{Grisha Perelman, in a few short papers -- whose complete details have
since been filled in by careful work of Morgan et al. -- made the Ricci flow
with surgery work. It appears that the complete Geometrization Conjecture,
and particularly the Poincar\'e Conjecture, is now settled in the
affirmative.}]
\section{Fundamentals}\label{sec:fundamentals}
\begin{definition} We use the standard notation for things like $\NN$ and
$\RR$ and $\CC$, but perhaps we should mention:
$$
\begin{array}{rl}
\textrm{for $n\in\NN\cup\{0\}$, }&S^n=\{x\in\RR^{n+1}\mid\abs{x}=1\}\\
&C^k=\{\textrm{functions (on an appropriate domain) which have $k$
continuous derivatives}\}\\
&C^\infty=\{\textrm{``smooth'' functions}\} = \cap_{k=1}^\infty C^k\\
&F_{abc\dots} = \textrm{the free group on generators\ }a,b,c,\dots
\textrm{\ (\emph{i.e.,} those generators and no relations)}\\
\textrm{for \emph{operators} $A$ and $B$, }&[A,B] =
A\circ B - B\circ A \textrm{\ \ (but beware similar notation in homotopy
theory, see \S\ref{sec:homotopy}, below)}
\end{array}
$$
\end{definition}
\begin{definition}
We say that a group $G$ \emph{acts on} a space $X$ if there is a
map $G\times X\to X:(g,x)\mapsto g\cdot x$ satisfying $(gh)\cdot x =
g\cdot(h\cdot x)$. The \emph{quotient} $G\backslash X$ is the collection of
equivalence classes of points in $X$, where two points are equivalent
if there is some group element in $G$ which maps one to the other.
\end{definition}
\section{Point-Set Topology}\label{sec:pstop}
\begin{definition} A \emph{topological space} is a set $X$ together
with a collection $\Oo$ of subsets of $X$ satisfying: \emph{(i)}
$\emptyset,X\in\Oo$; \emph{(ii)} arbitrary unions of elements in $\Oo$ are
again in $\Oo$; and \emph{(iii)} finite intersections of elements in $\Oo$
are again in $\Oo$. Elements of $\Oo$ are called \emph{open sets}, while
complements of open sets are called \emph{closed sets}.
\end{definition}
\begin{sidebar}
The idea of a topology is that these open sets are \emph{neighborhoods} of
each of their points; that is, the collection of all open sets $O_x$ containing
some point $x\in X$ is the collection of all ``sufficiently small
neighborhoods of $x$''=``sets of points sufficiently close to $x$'', however
\emph{in a way that doesn't require a definition of distance}. Hence it makes
sense to define
\end{sidebar}
\begin{definition}
A \emph{continuous map} $f:X\to Y$ between topological spaces is a map for
which the inverse image of any open set (in $Y$) is open (in $X$).
\end{definition}
Many topologies are \emph{metrizable}, in that there is a metric (in the
sense, at the moment, of a function which measures the distances between
pairs of points) for which the open sets are simply unions of open balls --
where an open ball in a metric space is the collection of those points
closer to some fixed point than a given number. All topological spaces we
shall deal with today are metrizable.
\begin{definition}
A nonempty subset $A$ of a topological space $(X,\Oo)$ is said to be a
\emph{connected component} if it is both open and closed. A topological
space which consists of only one connected component is said to be
\emph{connected}
\end{definition}
\begin{definition}
A subset $K$ of a topological space $(X,\Oo)$ is said to be \emph{compact}
if any open cover of $K$ has a finite subcover. For metrizable spaces, this
is equivalent to: any sequence has a convergent subsequence.
\end{definition}
\begin{definition}
A \emph{homeomorphism} between topological spaces is a continuous bijection
with continuous inverse. Two spaces are \emph{homeomorphic} if there exists
a homeomorphism between them.
\end{definition}
\begin{exercise}
Find topologies $\Oo_1$ and $\Oo_2$ on the set $\RR$ such that
the only continuous functions from $(\RR,\Oo_1)$ to $(\RR,\Oo_2)$
are constant. Find other topologies so that {\bf all} functions
are continuous. [\emph{Hint:} try very ``coarse'' or ``fine'' topologies
-- that is, ones with very few or very many open sets.]
\end{exercise}
\begin{exercise}
Fix $n\in\NN$. On $\CC^n$ consider the collection $\Cc$ of sets which are
the zero sets of complex polynomials in $n$ variables. Does this define
the closed sets of some topology on $\CC^n$ -- \emph{i.e.,} is
$\Oo=\{A^c\mid A\in\Cc\}$ a topology? [\emph{Hint:} if it were a topology,
it would be called the \emph{Zariski topology}.] On $\CC$ or $\CC^2$, for
example, does this topology agree with the usual one? If so, exhibit a
homeomorphism; if not, show some qualitatively different behavior.
\end{exercise}
\begin{exercise}\label{ex:topEC}
\emph{EXTRA CREDIT:} Show that $\RR^n$ is never homeomorphic to $\RR^m$ if
$n\neq m$. Is there ever a bijection between Euclidean spaces of different
dimensions? [\emph{Research:} ``space-filling curves''.]
\end{exercise}
\section{Homotopy}\label{sec:homotopy}
\begin{definition}
Two continuous maps between topological spaces $f_0,f_1:X\to Y$ are
said to be \emph{homotopic}, written $f_0\simeq f_1$, if there exists
a continuous map $F:[0,1]\times X\to Y$ such that $F(0,t)=f_0(t)$ and
$F(1,t)=f_1(t)$. The collection of homotopy classes of maps from $X$ to
$Y$ is written $[X,Y]$.
\end{definition}
\begin{definition}For a topological space $X$, $\pi_n(X)=[S^n,X]$.
\end{definition}
If $n=0$: $\pi_0(X)$ is the collection of \emph{path components} of $X$.
Usually -- certainly for all the spaces we shall deal with today -- this
is the same as the set of connected components of $X$.
If $n>0$: $\pi_n(X)$ is a group under \emph{concatenation} of paths
(\emph{i.e.}, do one path then the other), and therefore $\pi_n(X)$ is
also called the \emph{$n$th homotopy group of $X$}. A special case:
\begin{definition}
The \emph{fundamental group} of a topological space $X$ is $\pi_1(X)$.
If $\pi_1(X)$ is the trivial group, we say $S$ is \emph{simply connected}.
\end{definition}
Hence a simply connected space is one for which any closed loop can be
continuously contracted to a point.
Note that if $n>1$, $\pi_n(X)$ is Abelian, while the fundamental group is
often not so. In fact, the Abelianization of the fundamental group is (for
the spaces we shall deal with today) the same as the \emph{$1^\text{st}$
homology group}.
\begin{definition}
We say two topological spaces $X$ and $Y$ have the same \emph{homotopy type}
if there exist maps $f:X\to Y$ and $g:Y\to X$ which are inverses up to
homotopy, in the sense that $f\circ g\simeq\text{Id}_Y$ and
$g\circ f\simeq\text{Id}_X$; such a map $f$ (and likewise $g$) is called a
\emph{homotopy equivalence}.
\end{definition}
Note that for many spaces (\emph{e.g.}, all \emph{CW}-complexes, so all spaces
we shall deal with today), a map which induces an isomorphism of all homotopy
groups is automatically a homotopy equivalence. Hence one often talks of
spaces which are ``homotopy $n$-spheres'', that is, spaces with the homotopy
groups of an $n$-sphere. Similarly, there is a weaker notion of a ``homology
sphere'', being a space which has the \emph{homology groups} of a sphere.
But the definition of homology groups is quite complicated (and we don't
really need it today), so we won't give it in its full glory here. We shall,
however, talk a bit about \emph{(de Rham) cohomology} a bit -- see
\S\ref{sec:coh}, below.
\begin{exercise}
Exhibit a homotopy equivalence between $\RR^n$ and $\RR^m$.
\end{exercise}
\begin{exercise}
What is the fundamental group of the M\"obius strip? of the projective
plane? For $j,k=1,2,3$, compute $\pi_j(S^k)$ -- work them all out, there
are some surprises; some of the generators are famous geometric constructions
[\emph{Research:} ``the Hopf fibration'']; some of the proofs of rather
intuitively obvious parts of this exercise are surprisingly tricky
(\emph{e.g.,} when computing $\pi_1(S^2)$, what about the situation mentioned
in the above Exercise \ref{ex:topEC}?).
\end{exercise}
\begin{exercise}
\emph{EXTRA CREDIT (and almost certainly a Fields Medal):} For any $j,k\in\NN$,
compute $\pi_j(S^k)$.
\end{exercise}
\begin{definition}
A \emph{covering} $p:\tilde{X}\to X$ of a topological space is a surjective
continuous map with the property that every point $x\in X$ has a neighborhood
$U_x$ such that $p^{-1}(U_x)$ is a disjoint union of sets each of which is
mapped homeomorphically onto $U_x$ by $p$. A \emph{universal cover} of $X$
is a covering whose total space $\tilde{X}$ is simply connected.
\end{definition}
\section{Manifolds}\label{sec:manifolds}
\begin{definition}
A \emph{smooth $n$-manifold} is a topological space $X$, an a open cover
$\{U_\alpha\}_{\alpha\in A}$ of $X$, and maps
$\{\varphi_\alpha:U_\alpha\to\RR^n\}_{\alpha\in A}$ which are homeomorphisms
onto their images. The maps are required to satisfy a compatibility
condition as follows: for all $\alpha,\beta\in A$, let
$U_{\alpha\beta}=U_\alpha\cap U_\beta$; consider
$$
\varphi_\beta\circ \varphi_\alpha^{-1}:\varphi_\alpha(U_{\alpha\beta})\to
\varphi_\beta(U_{\alpha\beta})\qquad;
$$
this map must be \underline{smooth}. The open sets $U_\alpha$ are called
\emph{coordinate patches} for the manifold.
\end{definition}
One can play this game in other categories: by relaxing the restriction on
the composite maps to be merely \underline{continuous}, one gets a
\emph{topological manifold}; requiring the composites to be
p\underline{iecewise linear}, one gets a \emph{PL manifold}; requiring
\underline{real anal}y\underline{tic} composites, one
gets a \emph{real analytics manifold}; \emph{etc.} Today we shall stick to
the smooth category.
\begin{sidebar}
The notion of a manifold is designed to incorporate the concept of a space
which is \emph{locally} Euclidean -- like enough to Euclidean space that,
for example, we can do calculus -- but \emph{globally} it may have much more
structure, indeed it may have non-trivial topology. The simplest way to do
calculus on a manifold starts with
\end{sidebar}
\begin{definition}
The \emph{tangent space} $T_xX$ at a point $x$ in a manifold $X$
consists of equivalence classes of curves passing through $x$,
where two curves are considered equivalent if they agree to first order
in any coordinate patch containing $x$.
\end{definition}
These tangent spaces have the structure of a vector space coming from that
on $\RR^n$. [As follows: translate the patch so that $x$ corresponds to
$0\in\RR^n$, then simply use the scalar multiplication and vector addition
of $\RR^n$ to get new curves from old; this passes to equivalence classes
of curves.] Putting together these tangent spaces at all points of $X$
gives
\begin{definition}
The \emph{tangent bundle} $TX$ of a manifold is the union
$\bigcup_{x\in X} T_xX$ of all the tangent spaces at the points of $X$;
the projection $p:TX\to X$ takes a vector $v\in T_xX$ to $x$. ($p:TX\to X$
is actually a vector bundle over $X$.) A \emph{vector field on $X$} is a
\emph{section} of the tangent bundle, that is, it is a function $A:X\to TX$
with the property that $p(A(x))=x$; \emph{i.e.,} for each $x\in X$, $A(x)$
is a vector in the tangent space $T_xX$.
\end{definition}
Vector fields allow us to do (a little bit of) calculus on manifolds:
\begin{definition}
If $A$ is a vector field and $f$ a function on a manifold $X$, then
$A\cdot f$ is the function defined at $x\in X$ by
$(A\cdot f)(x)=
\left.\dfrac{\partial(f\circ\gamma)}{\partial t}\right|_{t=0}$ if
$\gamma(t)$ is a curve in the equivalence class defining $A(x)$ for which
$\gamma(0)=x$.
\end{definition}
\begin{references}
A nice basic book is \cite{Warner}. A book which emphasizes the
\emph{calculus} aspect is the appropriately named \cite{SpivakCoM}.
\end{references}
\section{De Rham Cohomology}\label{sec:coh}
*\emph{This section is optional (and assumes more linear algebra background
than the rest).}
\begin{definition} We need additional notation:
$$
\begin{array}{rl}
\textrm{for a vector space $V$,}&\Lambda^p(V) = \{p-\textrm{multilinear,
skew-symmetric maps $V\to\RR$}\}\\
\textrm{[Note:}&\Lambda^0(V) = \RR\textrm{\ \ (by convention)}\\
\textrm{and}&\Lambda^1(V) = V^*\textrm{\ \ (the dual space)]}\\
\textrm{for $\alpha\in\Lambda^p(V)$ and $\beta\in\Lambda^q(V)$, }
&\alpha\wedge\beta\in\Lambda^{p+q}(V)\textrm{\ is defined by }
(\alpha\wedge\beta)(v_1,\dots,v_{p+q})=\\
&\textrm{\ \ \ }=
\sum\limits_{\sigma\in S_{p+q}}\sgn(\sigma)\alpha(v_{\sigma(1)},\dots,
v_{\sigma(p)})\cdot\beta(v_{\sigma(p+1)},\dots,v_{\sigma(p+q)})
\end{array}
$$
\end{definition}
Now,
\begin{definition}
The \emph{bundle $\Lambda^p(X)$ of $p$-forms on a manifold $X$} is the
vector bundle built out of the tangent bundle $TX$ by replacing each
fiber $T_xX$ by $\Lambda^p(T_xX)$. The (vector) space of sections
of this bundle is denoted $\Omega^p(X)$ and called the space of
\emph{$p$-forms on $X$}.
\end{definition}
\begin{definition}
On a smooth manifold $X$, there is a (first order differential) operator
$d:\Omega^p(X)\to\Omega^{p+1}(X)$ defined by $df(A)=A\cdot f$ for a
function $f\in\Omega^0(X)$ and vector field $A$, and
$d(\alpha\wedge\beta)=(d\alpha)\wedge\beta+ (-1)^p\beta\wedge(d\alpha)$
for $\alpha\in\Omega^p(X)$ and $\beta\in\Omega^q(X)$, called the
\emph{exterior derivative}.
\end{definition}
More importantly, by the linear algebra of skew forms and the equality
of mixed partials, $d^2=0$.
\begin{exercise}
Prove that $d^2=0$.
\end{exercise}
Hence it makes sense to define
\begin{definition}
The \emph{de Rham cohomology in dimension $p$ of a manifold $X$} is
the vector space
$$
H^p_{DR}(X) = \left.\mathrm{ker}(d\right|_{\Omega^p(X)})/\left.\mathrm{Im}(d\right|_{\Omega^{p-1}(X)})\qquad .
$$
The corresponding dimension $b_p=\mathrm{dim}(H^p_{DR}(X))$ is called the
\emph{$p$th Betti number of $X$}, and the \emph{Euler characteristic}
of an $n$-manifold $X$ is then defined as
$$
\chi(X) = \sum_{p=0}^n (-1)^p b_p\qquad .
$$
\end{definition}
\begin{sidebar}
Once we have a Riemannian metric on a compact, connected smooth manifold $X$
(see \S\ref{sec:riemgeo}, below), there is defined an adjoint operator $d^*$
and a corresponding ``Laplacian $\Delta = d\! d^* + d^*\! d$ on $p$-forms'',
the kernel of which is the space $\Hh^p(X)$ of \emph{harmonic $p$-forms}.
Then ``Hodge Theory'' tells us that $\Hh^p(X)\cong H^p_{DR}(X)$.
\end{sidebar}
\begin{exercise}
Prove, starting from the above definition, that the $0^\textrm{th}$ de Rham
cohomology of a manifold $X$ obeys $H^0_{DR}(X)=\RR^k$, where $k$ is
the number of connected components in $X$.
\end{exercise}
\begin{exercise}
\emph{EXTRA CREDIT + Research:} Read about ``the Mayer-Vietoris exact
sequence'' (\emph{e.g.,} in \cite{BottTu}) and then go compute the
de Rham cohomology of all compact $2$-manifolds, triangulate these
surfaces and compute the traditional Euler characteristic, showing it
equals the de Rham version defined above.
\end{exercise}
\begin{references}
Bott's book \cite{BottTu} is quite beautiful. \cite{Warner} has good
coverage of Hodge theory.
\end{references}
\section{Riemannian Geometry}\label{sec:riemgeo}
\begin{sidebar}
On an $n$-manifold $X$ we have the ``distanceless notion of nearness''
inherent in its underlying topology, and some simple calculus from
the local Euclidean structure. To go on we want actual distances,
which we build up from a smoothly varying inner product on the tangent
spaces of $X$. From this will follow a (distance) metric, and a more refined
differentiation operator (the Levi-Civita connection), which in turn yields
(one version of) a notion of ``straight lines'', parallel transport and even
\emph{curvature}.
\end{sidebar}
\begin{definition}
A smooth choice of inner products on the tangent spaces of a manifold
$X$ is called a \emph{Riemannian metric}; so given vector fields $A$
and $B$, we write $g(A,B)$ for the function which at $x\in X$ is the
inner product of $A(x)$ with $B(x)$ (and the smoothness of $g$ simply
means that this function is smooth for smooth vector fields $A$ and $B$).
A manifold with Riemannian metric is called a \emph{Riemannian manifold}.
\end{definition}
\begin{definition}
Given a (piecewise $C^1$) curve $\alpha:[0,1]\to X$ in a Riemannian manifold
$X$, we define the \emph{length} of $\alpha$ to be
$$
\Ll(\alpha)=\int_0^1 \left(\left.g\right|_{\alpha(t)}
\left(\frac{\partial\alpha}{\partial t},
\frac{\partial\alpha}{\partial t}\right)\right)^\frac12\ dt\quad .
$$
For points $x_1$ and $x_2$ in a Riemannian manifold, we define the
\emph{distance between $x_1$ and $x_2$} to be
$$
d(x_1,x_2) = \inf_{\alpha:x_1\leadsto x_2} \Ll(\alpha)
$$
where the infimum is over all (piecewise $C^1$) paths $\alpha$ with
$\alpha(0)=x_1$ and $\alpha(1)=x_2$.
\end{definition}
So, apparently, an inner product in the fibers of the tangent bundle
gives a metric (in the sense of distance function) on the manifold
itself. This gives rise then to generalizations of ``straight lines'',
to wit
\begin{definition}
A curve $\alpha:[01]\to X$ in a Riemannian manifold is called a
\emph{geodesic} if for all sufficiently close pairs $t_1,t_2\in[0,1]$
we have $d(\alpha(t_1),\alpha(t_2))=\Ll(\left.\alpha\right|_{[t_1,_2]})$.
\end{definition}
That is, a geodesic is a locally length-minimizing curve. An equivalent
definition of geodesics comes instead from a special notion of
differentiation which any manifold can be given:
\begin{definition}
A \emph{connection on $TX$} is a (differential) operator which takes
two vector fields $A$ and $B$ and returns another vector field
denoted $\nabla_A B$ which is linear over $\mathrm{C}^\infty(X)$ in
the $A$ and over $\RR$ in the $B$, and satisfies
$\nabla_A(fB)=(A\cdot f)B+f\nabla_A B$. The \emph{torsion} of such
a connection is defined to be $T_\nabla(A,B)=\nabla_A B-\nabla_B A - [A,B]$,
where, as usual, $[A,B] = AB-BA$ for any operators $A$ and $B$.
\end{definition}
There is a particularly nice connection in the situation we are most
interested in:
\begin{definition}
A Riemannian manifold $(X,g)$ has a unique connection, called the
\emph{Levi-Civita connection} with vanishing torsion and for which
$\nabla g = 0$ (here we must extend $\nabla$ in a straightforward way
to operate on objects like $g$; equivalently,
$A\cdot g(B,C) = g(\nabla_A B,C) + g(B,\nabla_A C)$). We shall
\end{definition}
We shall use the Levi-Civita without further ado whenever we need a
connection on a Riemannian manifold. For example,
\begin{fact}
A curve $\alpha$ in a Riemannian manifold $X$ is a geodesic if and only
if $\nabla_{\dot{\alpha}(t)}\dot{\alpha}(t) = 0$ [we say the tangent
vector is \emph{parallel} along $\alpha$]. At any $x$ on a complete
Riemannian manifold $X$, there is a map which takes $v\in T_xX$ to
$\alpha_v(1)\in X$, where $\alpha$ is a geodesic with $\dot{\alpha}(0)=v$;
this map is called the \emph{exponential map} and is a local diffeomorphism.
\end{fact}
\begin{definition}
Given a Riemannian manifold $(X,g)$, a \emph{conformal change of metric}
is a metric $e^fg$ on $X$, where $f$ is some smooth function on $X$. Two
metrics which are related by such a change are called \emph{conformal},
and a \emph{conformal structure} is a conformal class of metrics.
\end{definition}
\begin{definition}
A diffeomorphism of Riemannian manifolds which preserves the respective
metrics is called an \emph{isometry}. The \emph{isometry group} of a
Riemannian manifold is the group of self-diffeomorphisms which are also
isometries.
\end{definition}
An important invariant of a Riemannian manifold is
\begin{definition}
The \emph{curvature tensor of a Riemannian manifold $X$} is (the
endomorphism-valued $2$-form) defined as the vector field
$R(A,B)C = \nabla_{[A,B]} C - [\nabla_A,\nabla_B] C$ for vector
fields $A$, $B$ and $C$ on $X$. The \emph{sectional curvature} of a
$2$-plane $\sigma\subseteq T_xX$ with basis $\{A, B\}$ is
$K(\sigma)=g(R(A,B)A,B)/(g(A,A)g(B,B)-g(A,B)^2)$. The \emph{Ricci
curvature} of $X$ is the $2$-tensor which on vector fields $A$ and
$B$ gives the function $r(A,B)=\mathrm{tr}(C\to R(A,C)B)$. The
\emph{scalar curvature} is the function $s$ on $X$ defined by
$s=\mathrm{tr}_g(r)$.
\end{definition}
\section{Dimension Two}\label{sec:dimtwo}
In this section, $\Sigma$ will be a compact, connected manifold of
dimension two.
\begin{definition}
$\Sigma$ is said to be \emph{orientable} if is possible to choose
a global non-vanishing $2$-form on $\Sigma$. This amounts to being
able to make a consistent choice of the notion of ``counterclockwise
rotation'' in each of the tangent spaces of $\Sigma$.
\end{definition}
\begin{sidebar}
A conformal class of Riemannian metric on an orientable $\Sigma$
allows us to identify each tangent space, which can already be
(non-canonically) identified with $\RR^2$, in fact with $\CC$:
an orientation together conformal structure lets us construct an
operator in each tangent space which is ``rotation by angle $\pi/2$
in a counterclockwise direction -- which is exactly what we need
to be the operation of multiplication by $i$. This endows $\Sigma$
with a \emph{complex structure}.
\begin{references}
A nice place to start in studying complex manifolds is the
(oddly, modestly named) \cite{Chern}.
\end{references}
\end{sidebar}
\begin{definition}
On our $2$-dimensional $\Sigma$, the sectional curvature, defined above,
has only one value at each point $x\in\Sigma$ [$T_xX$ is \emph{itself}
a $2$-plane], hence defines a function which is called the \emph{Gaussian
curvature} and written $K$ as well.
\end{definition}
The following is a very classical theorem (known to Gauss himself!),
which can be proven in a number of ways.
\begin{GaussBonnet}
For an arbitrary metric on our $\Sigma$,
$$
\int_\Sigma K\ d\textrm{vol} = 2\pi\chi(X)=2\pi(2-2g)
$$
where $g$ is the genus of $\Sigma$ and $\chi(X)$ its Euler characteristic,
defined as above in \S\ref{sec:coh} or in classical terms as
$V-E+F$ for a triangulation.
\end{GaussBonnet}
A nice application of non-linear global analysis gives:
\begin{fact}
Any metric on our $\Sigma$ can be conformally changed to have
constant scalar (Gaussian) curvature of $-1$, $0$ or $1$, depending
upon the sign constraint imposed by the Gauss-Bonnet Theorem and $\chi(X)$.
\end{fact}
It follows that a $\Sigma$ topologically the sphere can be given the
metric of the round $2$-sphere; a $\Sigma$ topologically the torus can
be given a flat metric and hence has $\RR^2$ as universal cover; and
a $\Sigma$ of genus $g\ge2$ has a constant negative curvature metric,
with universal cover isometric to
\begin{definition}
\emph{$2$-dimensional hyperbolic space $\Hh^2$} is the manifold diffeomorphic
to $\RR^2$ with either of the following realizations:
\begin{enumerate}
\item $\Hh^2 = \{(x,y)\in\RR^2\mid y>0\}$ with the Riemannian metric at
$(x,y)$ being the usual metric of $\RR^2$ divided by $y$.
\item $\Hh^2 = \{(x,y)\in\RR^2\mid x^2+y^2<1\}$ with the Riemannian metric
at $(x,y)$ being the usual metric of $\RR^2$ divided by $(1-x^2-y^2)$.
\end{enumerate}
\end{definition}
\begin{fact}
The isometry group of $\Hh^2$ can be identified with $PSL(2,\RR)$ acting
by fractional linear transformations.
\end{fact}
\section{Dimension Three}\label{sec:dimthree}
In this section, $X$ will be a compact, connected Riemannian manifold of
dimension three.
\begin{definition}
A \emph{connected sum} of two $3$-manifolds $X_1$ and $X_2$ is a
third $3$-manifold formed by puncturing $X_1$ and $X_2$ and then
identifying small neighborhoods of the punctures (each of which is
essentially a punctured ball in $\RR^3$) in the separate pieces.
A manifold is \emph{prime} if there is no non-trivial way to write it
as a connected sum. [A \emph{trivial connected sum} is a connected sum
of some manifold with $S^3$.]
\end{definition}
\begin{definition}
A $3$-manifold is \emph{irreducible} if every embedded $S^2$ bounds a $3$-ball.
\end{definition}
\begin{definition}
A torus $T$ embedded in a $3$-manifold $X$ is called \emph{incompressible}
if the inclusion map induces an injection on fundamental groups $\pi_1$;
\emph{i.e.,} if every loop which is homotopically nontrivial in $T$
remains nontrivial even up to homotopy in the ambient $X$.
\end{definition}
\begin{fact}
Every compact, connected irreducible $3$-manifold admits a (finite!) maximal
connected sum decomposition into irreducible pieces. [This is the
\emph{Sphere (or Prime) Decomposition}, due to Kneser and Milnor.]
Every compact, connected irreducible $3$-manifold admits a finite maximal
collection of disjoint incompressible tori. [This is \emph{Torus
Decomposition} of Jaco-Shalen.]
\end{fact}
\begin{ThurstonConj}
Decompose at compact, connected $3$-manifold by the Sphere and Torus
Decompositions. Then the resulting fragments each admit a finite volume
geometry of one of the following eight types:
\begin{enumerate}
\item Euclidean (flat) geometry (so the fragment is a quotient of $\RR^3$
by a group of rigid motions);
\item hyperbolic geometry (so constant negative curvature and the fragment
is isometric to a quotient of $3$-dimensional hyperbolic space by the
action of a discrete group of isometries ($PSL(2,\CC)$));
\item spherical geometry (so constant positive curvature and the fragment
is isometric to a quotient of $S^3$ by a discrete subgroup of $O(3)$);
\item the geometry of $S^2\times\RR$;
\item the geometry of $\Hh^2\times\RR$;
\item the geometry of the universal cover of the group $SL(2,\RR)$;
\item Nil geometry -- the geometry of the Heisenberg group (the group
of matrices of the form
$\left(\begin{matrix}1&x&y\\0&1&z\\0&0&1\end{matrix}\right)$); or
\item Sol geometry (the geometry of the semidirect product of $\RR$ with
$\RR^2$, where the former operates on the latter by
$(z,(x,y))\mapsto(e^zx,e^zy)$).
\end{enumerate}
\end{ThurstonConj}
\begin{sidebar}
Actually, the torus decomposition of the Conjecture is slightly
different from Jaco-Shalen's, in certain easily identifiable cases.
\end{sidebar}
\begin{references}
The book \cite{Thurston} by Wild Bill Thurston himself (with help from
Silvio Levy) puts a lot of this together, also with much background
in $2$-manifolds and connections to other areas.
\end{references}
\begin{PerelmanProof}
Perelman's proof is outlined in the three e-prints \cite{Perelman1},
\cite{Perelman2} and \cite{Perelman3}. More recently, Morgan and Tian
\cite{MorganTian} and Kleiner and Lott \cite{KleinerLott} have posted
e-prints, based on Perelman, with many more details. The basic
strucutre of the proof is to use the \emph{Ricci flow}, which is a
parabolic equation for the metric on the manifold, and to correct by
surgery certain catastrophes which occur in the flow.
\end{PerelmanProof}
\section{A Guide to (some) Relevant Literature\\
(or: some of my favorite books)}\label{sec:litguide}
In addition to what was mentioned in-line above, some literature to
look at would be:
\begin{itemize}
\item A very gentle introduction to some of the basic topology and geometry
is \cite{SingerThorpe}.
\item \cite{Katok} is a very nice (and easy) introduction to $\Hh^2$, its
isometry group, and subgroups thereof.
\item Doing topology in the context of differential geometry it makes sense
to look at Milnor's beautiful books \cite{MilnorTopology} and
\cite{MilnorMorse}.
\item There are many good books on basic differential geometry, beside the
one mentioned above -- for example: \cite{Lang}, \cite{SpivakDG} and
\cite{SpivakDGII} (the latter two are know to students as ``All the way
with Gauss--Bonnet'').
\item If you want to go on to yet higher dimensions, see
\cite{DonaldsonKronheimer} and \cite{FreedUhlenbeck}, among many others.
\item The serious analytic aspects of various curvature questions are
addressed in \cite{Kazdan}, while the serious aspects of Ricci curvature
are in \cite{Besse} (the latter being a book by a nonexistent person, oddly
enough).
\item Should any specific aspect of this area particularly interest you,
and should a comfortable and appropriate reference not be already mentioned
here, please contact me by e-mail and I would be happy to suggest many more
relevant sources.
\end{itemize}
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