Include a brief index

Signed-off-by: Marcello Seri <[email protected]>

Author: mseri

1-manifolds.tex

@@ -106,7 +106,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 	Show that the topological space defined on $\R^n$ is Hausdorff and second countable.
 \end{exercise}
 
-\begin{definition}[Topological manifold]
+\begin{definition}[Topological manifold]\idxdef{Topological manifold}
 	A topological space\sidenote[][-1.5em]{From now on, if we say that $X$ is a topological space we are implying that there is a topology $\cT$ defined on $X$.} $M$ is a \emph{topological manifold} of dimension $n$, or topological $n$-manifold, if it has the following properties:
 	\marginnote[-0.5em]{Note that the finite dimensionality is a somewhat artificial restriction: manifolds can be infinitely dimensional~\cite{book:lang:infinite}. For example, the space of continuous functions between manifolds is a so-called infinite-dimensional Banach manifold.\vspace{1em}}
 	\begin{enumerate}[(i)]
@@ -190,7 +190,7 @@ \section{Differentiable manifolds}
 	We say that $f$ is \emph{smooth}, or of class $C^\infty$, if it is of class $C^r$ for every $r \geq 1$.
 \end{definition}
 
-\begin{theorem}[Chain rule]\label{thm:chainrule}
+\begin{theorem}[Chain rule]\label{thm:chainrule}\idxthm{Chain rule}
 	Let $U\subseteq\R^n$ and $V\subseteq\R^k$ be open sets and $f: U \to \R^k$, $g: V\to\R^m$ two continuously differentiable functions such that $f(U)\subseteq V$.
 	Then, the following holds.
 	\begin{enumerate}[(i)]
@@ -243,7 +243,7 @@ \section{Differentiable manifolds}
 
 With these at hand, let's jump into the definition of smooth manifolds.
 
-\begin{definition}\label{def:cratlas}
+\begin{definition}\label{def:cratlas}\idxdef{Atlas}
 	A \emph{smooth atlas} is a collection
 	\begin{equation}
 		\cA = \{\varphi_\alpha: U_\alpha \to V_\alpha \;\mid\; \alpha\in A\}
@@ -257,15 +257,15 @@ \section{Differentiable manifolds}
 	Show that the equivalence of atlases is really an equivalence relation.
 \end{exercise}
 
-\begin{definition}\label{def:diffstr}
+\begin{definition}\label{def:diffstr}\idxdef{Smooth structure}
 	A \emph{differentiable structure}, or more precisely a \emph{smooth structure}, on a topological manifold is an equivalence class of smooth atlases.
 \end{definition}
 
 \begin{remark}
 	The union of all atlases in a differentiable structure is the \emph{unique} \emph{maximal} atlas in the equivalence class.\footnote{There is a one-to-one correspondence between differentiable structures and maximal differentiable atlases \cite[Proposition 1.17]{book:lee}: for convenience and to lighten the notation, from now on, we will always regard a differentiable structure as a differentiable maximal atlas without further comments.}
 \end{remark}
 
-\begin{definition}\label{def:diffmanifold}
+\begin{definition}\label{def:diffmanifold}\idxdef{Smooth manifold}
 	A \emph{smooth manifold} of dimension $n$ is a pair $(M, \cA)$ of a topological $n$-manifold $M$ and a smooth structure $\cA$ on $M$.
 	\marginnote[1em]{There are no preferred coordinate charts on a manifold: all coordinate systems compatible with the differentiable structure are on equal footing.}
 \end{definition}

10-integration.tex

@@ -8,7 +8,7 @@
 Not just that, you have seen already that in $\R^n$ there is a notion of standard orientation, but in other vector spaces we may need to make arbitrary choices.
 For manifolds, the situation is much more complicated: for example, on a M\"obius strip\footnote{Cf. Example~\ref{ex:mobius}.} it is impossible to make any such choice, it is non-orientable.
 
-\section{Orientation on vector spaces}
+\section{Orientation on vector spaces}\idxdef{Orientation}
 Let's proceed step by step by first revisiting some results from multivariable analysis. If you need a reference to review this material, you can refer to \cite[Chapter 6.2]{book:abrahammarsdenratiu} or \cite[Chapters 21.1-21.2]{book:tu}.
 
 \begin{definition}
@@ -80,7 +80,7 @@ \section{Orientation on manifolds}
 Differential $n$-forms, then, seem a reasonable concept to define a notion of orientation for a manifold, at least if we think about their pointwise meaning of assigning an orientation to each fiber of $TM$.
 
 \begin{definition}
-	A \emph{volume form}\footnote{Sometimes also called \emph{orientation form}.} on a $n$-dimensional smooth manifold $M$ is a $n$-form $\omega\in\Omega^n(M)$ such that $\omega(p) \neq 0$ for all $p\in M$.
+A \emph{volume form}\footnote{Sometimes also called \emph{orientation form}.}\idxdef{Volume form} on a $n$-dimensional smooth manifold $M$ is a $n$-form $\omega\in\Omega^n(M)$ such that $\omega(p) \neq 0$ for all $p\in M$.
 	We say that $M$ is \emph{orientable} if there exists a volume form on $M$.
 \end{definition}
 
@@ -608,7 +608,7 @@ \section{Stokes' Theorem}
 
 We are going to state the theorem, discuss some of its consequences and then give its proof.
 
-\begin{theorem}[Stokes' theorem]\label{thm:Stokes}
+\begin{theorem}[Stokes' theorem]\label{thm:Stokes}\idxthm{Stokes' theorem}
 	Let $M$ be an oriented $n$-manifold with boundary and let $\omega\in\Omega^{n-1}(M)$ be compactly supported.
 	Then,
 	\begin{equation}\label{eq:Stokes}

2-tangentbdl.tex

@@ -74,7 +74,7 @@ \section{Directional derivatives in Euclidean spaces}\label{sec:dd}
 	Define some nontrivial function $f: \R^3 \to \R^4$ and compute the 4 objects described above for that function.
 \end{exercise}
 
-\section{Germs and derivations}
+\section{Germs and derivations}\idxdef{Germ}\idxdef{Derivation}
 
 \newthought{To reach our goal, defining derivations on manifolds}, a direct extension of partial derivatives is not enough: we will need to introduce some more levels of abstraction.
 
@@ -482,7 +482,7 @@ \section{The differential of a smooth map}\label{sec:diffsmooth}
 
 It should not come a surprise that with the constructions developed so far not only do we have one such map, but we can directly relate it to a derivative.
 
-\begin{definition}\label{def:differentialMap}
+\begin{definition}\label{def:differentialMap}\idxdef{Differential (of a map)}
 	Let $F: M \to N$ be a smooth map between the smooth manifolds $M$ and $N$, and let $p\in M$.
 	The \emph{differential $d F_p$ of $F$ at $p$} is the map\footnote{In the differential geometry literature, the differential has many names: you can find it called \emph{tangent map}, \emph{total derivative} or \emph{derivative} of $F$.
 		Since it ``pushes'' tangent vectors forward from the domain manifold to the codomain, it is also called the \emph{pushforward}. If that was not enough, different authors use different notations for it: besides $dF_p(v)$, you can find $F_* v_p$, $F'(p)$, $T_pF$, $DF(p)[v]$ or variations thereof.}
@@ -737,7 +737,7 @@ \section{Tangent vectors as tangents to curves}
 	\label{fig:2_4-v_cur}
 \end{marginfigure}
 This means that we can actually give an alternative definition of $T_pM$ in terms of tangents to curves:
-\begin{definition}\label{def:tg:ascurvespeed}
+\begin{definition}\label{def:tg:ascurvespeed}\idxdef{Tangent vector}
 	A tangent vector at $p\in M$ is an equivalence class\footnote{Usually we say that two such equivalent curves have \emph{a first order contact at $p$}.} of smooth curves $\gamma:(-\epsilon, \epsilon)\to M$ such that $\gamma(0)=p$, where $\gamma\sim\delta$ if and only if $(\varphi\circ \gamma)'(0) = (\varphi\circ\delta)'(0)$ for some chart $\varphi$ centred about $p$ (see Lemma~\ref{lem:equiv_tg_curves}).
 \end{definition}
 
@@ -772,7 +772,7 @@ \section{Tangent vectors as tangents to curves}
 	\textit{\small Hint: use the definitions to rewrite the formula in different ways.}
 \end{exercise}
 
-\section{The tangent bundle}\label{sec:tangentbundle}
+\section{The tangent bundle}\label{sec:tangentbundle}\idxdef{Tangent bundle}
 
 Instead of working separately with the various tangent spaces, we can ``glue'' them together into a big manifold.
 

3-submanifolds.tex

@@ -26,12 +26,12 @@ \section{Inverse function theorem}
 The same applies to the inverse function theorem:
 compare the following statements.
 %
-\begin{theorem}[Inverse function theorem for $\R^n$]\label{thm:ift}
+\begin{theorem}[Inverse function theorem for $\R^n$]\label{thm:ift}\idxthm{Inverse function theorem ($\R^n$)}
 	Let $U\subset\R^n$ open and $f:U \to \R^n$ be a smooth map.
 	Assume that $f$ has maximal rank at some $x\in U$, then there exists an open neighbourhood $\Omega\subset U$ of $x$ such that  $f\big|_\Omega : \Omega \to f(\Omega)$ is a diffeomorphism.
 \end{theorem}
 
-\begin{theorem}[Inverse function theorem for manifolds]\label{thm:iftm}
+\begin{theorem}[Inverse function theorem for manifolds]\label{thm:iftm}\idxthm{Inverse function theorem (manifolds)}
 	Let $F:M\to N$ be a smooth function between smooth manifolds without boundary of the same dimension $n$.
 	Let $p\in M$ such that $F$ has maximal rank $n$ at $p$.
 	Then there exists an open neighbourhood $V$ of $p$ such that the restriction $F|_V : V\to F(V)$ is a diffeomorphism.
@@ -69,7 +69,7 @@ \section{Inverse function theorem}
 The tool to get there is the following, we will see more clearly the link with
 projections and inclusions in the next subsection.
 
-\begin{theorem}[Rank theorem]\label{thm:rank}
+\begin{theorem}[Rank theorem]\label{thm:rank}\idxthm{Rank theorem}
 	Let $F : M^m \to N^n$ be a smooth function between smooth manifolds without boundary\footnote{The theorem can be extended to allow $M$ with boundary and $N$ without boundary assuming $\ker dF_p \not\subseteq T_p\partial M$ but we will omit this case here to keep the discussion more contained and avoid unnecessary technicalities.}.
 	Assume that $F$ is of rank $k$ at all points $p\in M$.
 	Then, for all $p\in M$ there exist smooth charts $(U, \varphi)$ centred at $p$ and $(V, \psi)$ centred at $F(p)$ with $F(U)\subseteq V$, such that the coordinate representation of $F$ with respect to the charts $\varphi$ and $\psi$ has the form

4-vectorfields.tex

@@ -6,7 +6,7 @@ \section{Vector fields}
 
 \newthought{The step to abstract manifolds} is rather intuitive in this case: a vector field will be a map that, at each point of a manifold, picks a tangent vector at that point in a smooth way.
 
-\begin{definition}\label{def:vfield}
+\begin{definition}\label{def:vfield}\idxdef{Vector field}
 	A $C^k$-map $X: M \to TM$ with $\pi\circ X = \id_M$, or equivalently $X_p\in T_pM$ for all $p\in M$, is called \emph{$C^k$-vector field}. Here, the map $\pi:TM \to M$ is the projection that we defined when we introduced the tangent bundle $TM$ in Section~\ref{sec:tangentbundle}.
 	The vector field is smooth if it is a $C^k$-vector field\footnote{Observe that it needs to be $C^k$ w.r.t. both the differentiable structures on $M$ and $TM$.} for all $k\geq 1$.
 

5-vectorbdl.tex

@@ -2,7 +2,7 @@
 
 \section{Vector bundles}\label{sec:vec-bdls}
 
-\begin{definition}\label{def:vector_bundle}
+\begin{definition}\label{def:vector_bundle}\idxdef{Vector bundle}
 	A \emph{vector bundle of rank $r$} over a manifold $M$ is a manifold $E$ together with a smooth surjective map $\pi : E \to M$ such that, for all $p\in M$, the following properties hold:
 	\begin{enumerate}[(i)]
 		\item the \emph{fibre over $p$}, $E_p := \pi^{-1}(p)$, has the structure of vector space of dimension $r$;

6-cotangentbdl.tex

@@ -2,7 +2,7 @@ \section{The cotangent space}
 
 \newthought{The dual of a vector space} should be a well-known concept from linear algebra. We recall it here just for the sake of convenience.
 
-\begin{definition}
+\begin{definition}\idxdef{Dual space}
 	Let $V$ a vector space of dimension $n\in \N$.
 	Its \emph{dual space} $V^* := \cL(V, \R)$ is the $n$-dimensional real vector space of linear maps $\omega:V \to R$.
 	The elements of $V^*$ are usually called \emph{linear functionals} and for $\omega\in V^*$ and $v\in V$ it is common to write
@@ -21,7 +21,7 @@ \section{The cotangent space}
 
 In the previous chapter we defined the tangent space as a special vector space over each point in a manifold, which nicely fits in the requirements above.
 
-\begin{definition}
+\begin{definition}\idxdef{Cotangent space}
 	Let $M$ be a differentiable manifold and $p\in M$.
 	The dual space $T_p^*M := (T_pM)^*$ of the tangent space $T_pM$ is called the \emph{cotangent space} of $M$ at $p$.
 	The elements of $T^*_pM$ are called \emph{cotangent vectors}, \emph{covectors} or \emph{(differential) $1$-forms} at $p$.

8-differentialforms.tex

@@ -9,7 +9,7 @@
 
 \section{Differential forms}
 
-\begin{definition}
+\begin{definition}\idxdef{Alternating $k$-form}\idxdef{Differential form}
 	Let $V$ be a real $n$-dimensional vector space.
 	Let $S_k$ denote the \emph{symmetric group on $k$ elements}, that is, the group of permutations of the set $\{1,\ldots,k\}$.
 	Recall that for any permutation $\sigma\in S_k$, the \emph{sign of $\sigma$}, denoted $\sgn(\sigma)$, is equal to $+1$ if $\sigma$ is even\footnote{It can be written as a composition of an even number of transpositions} and $-1$ is $\sigma$ is odd\footnote{It can be written as a composition of an odd number of transpositions}.
@@ -41,7 +41,7 @@ \section{The exterior product}
 We also saw that tensors can be multiplied with the tensor product, which gives rise to a graded algebra on the free sum of tensor spaces.
 Since we are looking for an alternating product, the previous observation leads naturally to the following definition.
 
-\begin{definition}
+\begin{definition}\idxdef{Wedge product}
 	Let $V$ be a real $n$-dimensional vector space.
 	Given $k$ covectors $\omega^1, \ldots, \omega^k\in T_1^0(V)$, their \emph{wedge product} (or \emph{exterior product}) $\omega^1\wedge\ldots\wedge\omega^k$ is defined by
 	\begin{equation}

9-cohomology.tex

@@ -1,5 +1,5 @@
 \section{De Rham cohomology}
-\begin{definition}
+\begin{definition}\idxdef{de Rham cohomology}
 	We say that a smooth differential form $\omega\in\Omega^k(M)$ is \emph{closed} if $d\omega = 0$, and \emph{exact} if there exists a smooth $(k-1)$-form $\eta$ on $M$ such that $\omega = d\eta$.
 
 	The fact that $d\circ d = 0$ implies that every exact form is closed.
@@ -39,11 +39,11 @@ \section{De Rham cohomology}
 	If $F:M\to N$ is a smooth map, then $F^*$ induces a well-defined map $F^*:H_{\mathrm{dR}}^k(N) \to H_{\mathrm{dR}}^k(M)$ (denoted with the same symbol) via $[\omega]\mapsto[F^*\omega]$.
 \end{corollary}
 
-\section{Poincar\'e lemma}
+\section{Poincar\'e lemma}\idxthm{Poincar\'e lemma}
 Without further ado, let's look at a first version of Poincar\'e lemma on manifolds.
 As for all the local concepts we have seen so far, the proof will reduce the problem to a euclidean statement to which we will apply the Poincar\'e lemma that you have seen in multivariable calculus.
 
-\begin{theorem}
+\begin{theorem}\idxthm{Poincar\'e lemma}
 	Let $M$ be a smooth manifold and $\omega\in\Omega^k(M)$ closed, that is, $d\omega = 0$.
 	Let $U\subset M$ be open and diffeomorphic to a star-shaped domain\footnote{Cf. Lemma~\ref{lem:Taylor}.} of $\R^n$.
 	Then, there exists $\nu\in\Omega^{k-1}(U)$ such that $\omega|_U = d\nu$.
@@ -63,7 +63,7 @@ \section{Poincar\'e lemma}
 
 A crucial observation for our means is the \emph{homotopy invariance} of the de Rham cohomology, which is a scary sounding property which is formalised by the following statement.
 
-\begin{theorem}\label{thm:deRham-invariance}
+\begin{theorem}\label{thm:deRham-invariance}\idxthm{de Rham homotopy invariance}
 	Let $M$ be a smooth manifold and $[0,1]\times M$ the product manifold with boundary $(\{0\}\times M) \cup (\{1\}\times M) \cup ((0,1)\times\partial M)$.
 	Let $i_t : M \hookrightarrow [0,1]\times M$ be the injection $i_t(p) := (t,p)$ and $\pi : [0,1]\times M \to M$ the projection onto $M$.
 	Then, there is a linear map

aom.tex

@@ -125,6 +125,9 @@
 \usepackage{doi}
 \addbibresource{aom.bib}
 
+\usepackage{makeidx}
+\makeindex
+
 \newcommand{\cA}{\mathcal{A}}
 \newcommand{\cC}{\mathcal{C}}
 \newcommand{\cD}{\mathcal{D}}
@@ -197,6 +200,14 @@
 \newcommand{\LaTeXoverbrace}[2]{\overbrace{#1}^{#2}}
 }
 
+\newcommand{\idx}[1]{\index{#1}}
+\newcommand{\idxdef}[1]{\index{Definitions!#1}}
+\newcommand{\idxthm}[1]{\index{Theorems!#1}}
+% Index helpers
+\newcommand{\idx}[1]{\index{#1}}
+\newcommand{\idxdef}[1]{\index{Definitions!#1}}
+\newcommand{\idxthm}[1]{\index{Theorems!#1}}
+
 \title{Analysis\\ \noindent
   on\\ \noindent
   Manifolds
@@ -363,4 +374,6 @@ \chapter{Integration of forms}
 
 \printbibliography
 \addcontentsline{toc}{chapter}{Bibliography}
+\cleardoublepage
+\printindex
 \end{document}