Include a brief index

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

Author: mseri

1-manifolds.tex

@@ -55,7 +55,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 \newthought{Since to speak of continuity we need topological spaces}, it may be a good idea to remind you what they are and set some notation.
 I will be very brief: if you need a more extensive reminder, you can refer to Appendix A of either~\cite{book:tu} or~\cite{book:lee}.
 
-\begin{definition}
+\begin{definition}\idxdef{Topological space}
 	Let $X$ be some set and $\cT$ a set of subsets of $X$.
 	A pair $(X, \cT)$ is a \emph{topological space}\footnote{In such case the elements $O\in\cT$ of $\cT$ are all subsets of $X$ called \emph{open} subsets and $\cT$ is a \emph{topology} on $X$.} if
 	\begin{enumerate}[(i)]
@@ -67,7 +67,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 
 With topological spaces at hand, we can give a definition of continuity and introduce a way to compare topological spaces.
 
-\begin{definition}
+\begin{definition}\idxdef{Continuous map}\idxdef{Homeomorphism}
 	A map $f: X \to Y$ between two topological spaces $(X,\cT)$ and $(Y, \cU)$ is called:
 	\begin{itemize}
 		\item \emph{continuous} if $U\in\cU$ implies that $f^{-1}(U)\in\cT$, that is, preimages of open sets under $f$ are open;
@@ -80,7 +80,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 	\vspace{5pt}
 \end{marginfigure}
 
-\begin{definition}
+\begin{definition}\idxdef{Hausdorff space}
 	A topological space $(X, \cT)$ is \emph{Hausdorff} if every two distinct points admit disjoint open neighbourhoods. That is, for every pair $x\neq y$ of points in $X$, there exist open subsets $U_x, U_y\in\cT$ such that $x\in U_x$, $y\in U_y$ and $U_x \cap U_y = \emptyset$.
 \end{definition}
 
@@ -96,7 +96,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 
 Hausdorff spaces are still rather general: in particular, any metric space with the metric topology\footnote{Recall that in a metric space $X$ the \emph{metric topology} is defined in the following way: a set $U\subset X$ is called open if for any $x\in U$ there exists $\epsilon>0$ such that $U$ fully contains the ball of radius $\epsilon$ around $x$.} is Hausdorff.
 
-\begin{definition}
+\begin{definition}\idxdef{Second countable}
 	A topological space $(X, \cT)$ is \emph{second countable} if there exists a countable set $\cB\subset\cT$ such that any open set can be written as a union of sets in $\cB$.
 	In such case, $\cB$ is called a (countable) basis for the topology $\cT$.
 \end{definition}
@@ -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)]
@@ -116,7 +116,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 	\end{enumerate}
 \end{definition}
 
-\begin{notation}\label{def:coords_obj}
+\begin{notation}\label{def:coords_obj}\idxdef{Chart}
 	Reusing the notation of the definition above, we call \emph{(coordinate) chart} the pair $(U, \varphi)$ of a \emph{coordinate neighbourhood}\footnote{Or \emph{coordinate open set}} $U$ and an associated \emph{coordinate map}\footnote{Or \emph{coordinate system}.} $\varphi: U\to V$ onto an open subset $V=\varphi(U)\subseteq\R^n$ of $\R^n$.
 	Furthermore, we say that a chart is \emph{centred at $p\in U$} if $\varphi(p) = 0$.
 \end{notation}
@@ -184,13 +184,13 @@ \section{Differentiable manifolds}
 
 Before entering into the details of new definitions, let's recall what will be the most important tools throughout the rest of the course.
 
-\begin{definition}
+\begin{definition}\idxdef{Partition of unity}
 	A map $f: U \to V$ between open sets $U\subset\R^n$ and $V\subset\R^m$ is in $C^r(U,V)$ or \emph{of class $C^r$}, if it is continuously differentiable $r$-times.
 	It is called a $C^r$-\emph{diffeomorphism}\footnote{With this definition a homeomorphism is a $C^0$-diffeomorphism} if it is bijective and of class $C^r$ with inverse of class $C^r$.
 	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 (Euclidean)}
 	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}
@@ -883,7 +883,7 @@ \section{Partitions of unity}\label{sec:partition_of_unity}
 
 We are not there yet. To extend this result to our needs will need a new tool, which will be useful throughout the course and in many courses to come.
 
-\begin{definition}
+\begin{definition}\idxdef{Partition of unity}
 	Let $M$ be a smooth manifold. A \emph{partition of unity} is a collection $\{\rho_\alpha \mid \alpha\in A\}$ of functions $\rho_\alpha:M\to\R$ such that
 	\begin{enumerate}[(i)]
 		\item $0 \leq \rho_\alpha \leq 1$ for all $p\in M$ and $\alpha\in A$;
@@ -899,7 +899,7 @@ \section{Partitions of unity}\label{sec:partition_of_unity}
 
 Throughout the course we will be mostly interested in partitions of unity $\{\rho_\alpha \mid \alpha\in A\}$ which are \emph{subordinate} to an open cover $\{U_\alpha\mid\alpha\in A\}$, that is, such that $\supp_\alpha(\rho_\alpha) \subset U_\alpha$ for each $\alpha\in A$.
 
-\begin{theorem}\label{thm:partitionof1}
+\begin{theorem}\label{thm:partitionof1}\idxthm{Partition of unity (existence)}
 	\marginnote[1.5em]{We are going to omit the proof of this theorem, for its details you can refer to~\cite[Proposition 13.6]{book:tu} or~\cite[Theorem 2.23]{book:lee}.}
 	Let $M$ be a smooth manifold. For any open cover $\{U_\alpha\mid\alpha\in A\}$ of $M$, there exists a partition of unity $\{\rho_\alpha \mid \alpha\in A\}$ subordinate to $\{U_\alpha\mid\alpha\in A\}$.
 \end{theorem}
@@ -972,7 +972,7 @@ \section{Manifolds with boundary}\label{sec:mbnd}
 	Explicitly state the definitions above in the case of manifolds with boundary.
 \end{exercise}
 
-\begin{definition}\label{def:diffmanifoldwb}
+\begin{definition}\label{def:diffmanifoldwb}\idxdef{Manifold with boundary}
 	\marginnote{Remember that the differentiable structure is an equivalence class of smooth atlases.}
 	A \emph{smooth manifold with boundary} of dimension $n$ is a pair $(M, \cA)$ of a topological $n$-manifold with boundary $M$ and a smooth differentiable structure $\cA = \{(U_\alpha, \varphi_\alpha) \mid \alpha\in A\}$ on $M$.
 

10-integration.tex

@@ -11,7 +11,7 @@
 \section{Orientation on vector spaces}
 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}
+\begin{definition}\idxdef{Oriented manifold}
 	Let $V$ be a one-dimensional vector space. Then $V\setminus\{0\}$ has two components.
 	An \emph{orientation} of $V$ is a choice of one of these components, which one then labels as ``positive'' and ``negative''.
 	A \emph{positive basis} of $V$ then is a choice of any non-zero vector belonging to the positive component, while a \emph{negative basis} of $V$ is a choice of any non-zero vector belonging to the negative component.
@@ -25,7 +25,7 @@ \section{Orientation on vector spaces}
 If $V$ be a $n$-dimensional vector space, we know by Proposition~\ref{prop:dimLkV}, that $\Lambda^n(V)$ is a one-dimensional vector space.
 Moreover, if $\{e_1,\ldots,e_n\}$ is a basis for $V$, then $e^1\wedge\cdots\wedge e^n$ is a basis for $\Lambda^n(V)$.
 
-\begin{definition}
+\begin{definition}\idxdef{Orientation of a vector space}\idxdef{Positive orientation}
 	Let $V$ be a $n$-dimensional vector space.
 	An \emph{orientation} on $V$ is a choice of orientation\footnote{That is, we are talking about a representative from an equivalence class.} on the one-dimensional vector space $\Lambda^n(V)$.
 	Therefore there are exactly two orientations: we say that a basis $\{e_1,\ldots,e_n\}$ of $V$ is \emph{positive} (or positively oriented) if $e^1\wedge\cdots\wedge e^n$ is a positive basis of $\Lambda^n(V)$ and \emph{negative} (or negatively oriented) otherwise.
@@ -63,7 +63,7 @@ \section{Orientation on vector spaces}
 	Let $V$ be a $n$-dimensional vector space, prove that two nonzero $n$-forms on $V$ determine the same orientation if and only if each is a positive multiple of the other.
 \end{exercise}
 
-This allows to define an equivalence relation between orientations in terms of the nonzero elements in $\Lambda^n(V)$. We call these nonzero elements \emph{volume elements}.
+This allows to define an equivalence relation between orientations in terms of the nonzero elements in $\Lambda^n(V)$. We call these nonzero elements \emph{volume elements}\idxdef{Volume element} on $V$.
 
 Two volume elements $\omega_1, \omega_2$ are equivalent if there exists $c > 0$ such that $\omega_1 = c\, \omega_2$.
 Then, the previous exercise implies that the classes of equivalence $[\omega]$ of volume elements on $V$ uniquely determine orientations on $V$.
@@ -79,9 +79,9 @@ \section{Orientation on manifolds}
 Moreover, we can extend our point of view to the various tensor bundles over smooth manifold that we studied so far.
 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 $p\in M$.
-	We say that $M$ is \emph{orientable} if there exists a volume form on $M$.
+\begin{definition}\idxdef{Volume form}\idxdef{Orientation-preserving diffeomorphism}
+	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}\idxdef{Orientable manifold} if there exists a volume form on $M$.
 \end{definition}
 
 \begin{example}
@@ -162,7 +162,7 @@ \section{Orientation on manifolds}
 \end{proof}
 
 
-\begin{definition}
+\begin{definition}\idxdef{Oriented manifold}\idxdef{Orientation of a manifold}
 	A manifold $M$ with an oriented atlas is called \emph{oriented manifold}.
 	If an orientation exists, we call \emph{orientation} the equivalence class of atlases with the same orientation, i.e., the family of atlases whose union is still an oriented atlas.
 	Otherwise we say that the manifold is \emph{non-orientable}.
@@ -178,7 +178,7 @@ \section{Orientation on manifolds}
 
 In fact, we can go one step further and define what does it mean for a diffeomorphism to preserve or reverse orientation.
 
-\begin{definition}
+\begin{definition}\idxdef{Orientation-preserving diffeomorphism}
 	Let $(M, \omega)$ and $(N, \eta)$ be pairs of an oriented smooth manifold and its volume form.
 	A diffeomorphism $F: M \to N$ is called \emph{orientation preserving} if $F^* \eta$ is a volume form on $M$ with the same orientation as $\omega$.
 	We call it \emph{orientation reversing} if $F^*\eta$ has the opposite orientation as $\omega$.
@@ -374,7 +374,7 @@ \section{Integrals on manifolds}
 
 % TODO: properly explain how the change of variable in Euclidean space requires the absolute value of the Jacobian determinant while the transformation of a k-form only involves the determinant itself, and thus in our definition we need to take into account the orientation and compensate with a minus sign if necessary.
 
-\begin{definition}\label{def:intnform:chart}
+\begin{definition}\label{def:intnform:chart}\idxdef{Integral of n-forms supported on a single chart}
 	Let $M$ be a smooth $n$-manifold and $(U,\varphi)$ be a chart from an oriented atlas of $M$ with coordinates $(x^i)$.
 	If $\omega\in\Omega^n(M)$ is a $n$-form, $n > 0$, with compact support in $U$, we define the integral of $\omega$ as\sidenote[][-1em]{Recall that for a diffeomorphism $\phi$, $\phi_* = (\phi^{-1})^*$.}
 	\begin{equation}
@@ -418,7 +418,7 @@ \section{Integrals on manifolds}
 
 To be able to integrate charts which are not supported in the domain of a single chart, we now need the help of a partition of unity.
 
-\begin{definition}
+\begin{definition}\idxdef{Integral of n-forms on manifolds}
 	Let $M$ be a smooth oriented manifold and $\cA = \{(U_i,\varphi_i)\}$ a positively oriented atlas.
 	If $\omega \in \Omega^n(M)$ has compact support, then the \emph{integral of $\omega$} is defined as
 	\begin{equation}\label{eq:intnform}
@@ -449,7 +449,7 @@ \section{Integrals on manifolds}
 \end{proof}
 
 Which immediately implies the following nice result.
-\begin{theorem}[Global change of variables]\label{thm:gcv}
+\begin{theorem}[Global change of variables]\label{thm:gcv}\idxthm{Global change of variables}
 	Suppose $M$ and $N$ are oriented $n$-manifolds and $F:M\to N$ is an orientation preserving diffeomorphism.
 	If $\omega\in\Omega^n(N)$ has compact support, then $F^*\omega$ has compact support and the following holds
 	\begin{equation}
@@ -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.
 
@@ -203,7 +203,7 @@ \section{Germs and derivations}
 As you can see, working with equivalence classes is doable... but it is also unnecessarily cumbersome.
 As we did with atlases, we would like to get it over with.
 
-\begin{definition}
+\begin{definition}\idxdef{Derivation of functions at a point}
 	Let $M$ be a smooth manifold and $p\in M$.
 	Let $W\subseteq M$ be any neighbourhood of $p$.
 	\marginnote{We are still talking about derivations of functions at specific points, not to be confused with the derivations of the algebra $C^\infty(W)$ which we will introduce later and will be maps of the kind $C^\infty(W)\to C^\infty(W)$.}
@@ -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.}
@@ -502,7 +502,7 @@ \section{The differential of a smooth map}\label{sec:diffsmooth}
 	What is $d F_{(1,1,1)} \left(\frac{\partial}{\partial x^1} - 2 \frac{\partial}{\partial x^2}\right)$?
 \end{exercise}
 
-\begin{theorem}[The chain rule on manifolds]\label{thm:chainrule_mfld}
+\begin{theorem}[The chain rule on manifolds]\label{thm:chainrule_mfld}\idxthm{Chain rule (manifolds)}
 	Let $M, N, P$ be smooth manifolds and $F: M \to N$, $G: N\to P$ be two smooth maps. Then
 	\marginnote{The alternative $D$ notation, in this case, makes the relation to the usual chain rule even more evident: $D(G\circ F)(p) = DG(F(p))\circ DF(p)$.}
 	\begin{equation}
@@ -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.