ch 1: fix typos from Martijn up to page 17

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

1-manifolds.tex

@@ -89,7 +89,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 
 \begin{exercise}
   \begin{itemize}
-    \item Let $X$ be an arbitrary set. Show that $\cT:=\{\emptyset, X\}$ defines a topology on $X$, called the \emph{trivial topology}. Show that on $(X, \cT)$ any sequence in $X$ converges to every point of $X$, and every map from a topological space into $X$ is continuous.
+    \item Let $X$ be an arbitrary set. Show that $\cT:=\{\emptyset, X\}$ defines a topology on $X$, called the \emph{trivial topology}. Show that on $(X, \cT)$ any sequence in $X$ converges\footnote{We say that a sequence $\{x_n\}_{n\in\N}\subseteq X$ of points in some topological space $(X, \cT)$ \emph{converges} to a point $x\in X$ if and only if for each open neighbourhood $U\in\cT$ of $x$, there exists $N\in\N$ such that $x_n\in U$ for all $n\geq N$.} to every point of $X$, and every map from a topological space into $X$ is continuous.
     \item Let $X$ be an arbitrary set. Show that $\cT:=\mathcal{P}(X) := \{ A \mid A\subset X \}$, the powerset of $X$, defines a topology on $X$, called the \emph{discrete topology} in which every map $f : X \to Y$ to some other arbitrary topological space $(Y, \cU)$ is continuous.
   \end{itemize}
 \end{exercise}
@@ -116,8 +116,8 @@ \section{Topological manifolds}\label{sec:top_manifolds}
   \end{enumerate}
 \end{definition}
 
-\begin{notation}
-  Reusing the notation of the definition above, we call \emph{(coordinate) chart} the pair $(U, \varphi)$ of a \emph{coordinate neighbourhood} $U$ 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 $\R^n$.
+\begin{notation}\label{def:coords_obj}
+  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}
 
@@ -282,7 +282,7 @@ \section{Differentiable manifolds}
 
 \begin{exercise}
 	Let $\{(U_\alpha, \varphi_\alpha)\}$ be the maximal atlas on a manifold $M$.
-	For any open set $U\subseteq M$ and any point $p\in U$, prove the existence of a coordinate open set $U_\alpha$ such that $p\in U_\alpha\subset U$.
+  For any open set $U\subseteq M$ and any point $p\in U$, prove the existence of a coordinate open set\footnote{Cf. Notation~\ref{def:coords_obj}.} $U_\alpha$ such that $p\in U_\alpha\subset U$.
 \end{exercise}
 
 \begin{exercise}
@@ -388,7 +388,7 @@ \section{Differentiable manifolds}
 \end{example}
 
 \begin{example}[Product manifolds]\label{ex:pm}
-  Given two manifolds $(M_1, \cA_1)$ and $(M_2, \cA_2)$, we can define the \emph{product manifold} $M_1 \times M_2$ by equipping $M_1 \times M_2$ with the product topology\footnote{Open sets in the product are products of open sets from the respective topological spaces.} and covering the space with the atlas $\{ (U_1\times U_2, (\varphi_1, \varphi_2)) \;\mid\; (U_1, \varphi_1)\in\cA_1, (U_2, \varphi_2)\in \cA_2\}$.
+  Given two manifolds $(M_1, \cA_1)$ and $(M_2, \cA_2)$, we can define the \emph{product manifold} $M_1 \times M_2$ by equipping $M_1 \times M_2$ with the product topology\footnote{Open sets in the product are generated by products of open sets from the respective topological spaces.} and covering the space with the atlas $\{ (U_1\times U_2, (\varphi_1, \varphi_2)) \;\mid\; (U_1, \varphi_1)\in\cA_1, (U_2, \varphi_2)\in \cA_2\}$.
 \end{example}
 
 
@@ -483,7 +483,7 @@ \subsection{Quotient manifolds}\label{sec:quotient}
     \caption{The identification $\sim'$ of antipodal points maps the sphere to a disk. Embedding $\bS^n/\!\sim'$ in $\R^{n+1}$, one can define a map $\pi_D$ that projects the representative of $[x]$ in the north hemisphere orthogonally to the disk $D^n = \{x\in\R^{n+1} \mid \|x\|\leq 1, \; x^{n+1}=0\}$ (the equator is mapped to itself). }
   \end{marginfigure}
   There is a nice interpretation of this construction in terms of flattening spheres.
-  Observe that a line through the origin always intercepts a sphere $\bS^n$ at two antipodal points and, conversely, each pair of antipodal point determines a unique line through the center.
+  Observe that a line through the origin always intersects a sphere $\bS^n$ at two antipodal points and, conversely, each pair of antipodal points determines a unique line through the center.
   So we can define an equivalence relation on the sphere by identifying the antipodal points: given $x,y\in\bS^n$, $x\sim' y$ if and only if $x = \pm y$.
   This leads to the bijection $\RP^n \simeq \bS^n/\!\sim'$.
   Note that by gluing antipodal points, we are identifying the north and south hemispheres, thus essentially flattening the sphere to a disk.
@@ -501,7 +501,7 @@ \subsection{Quotient manifolds}\label{sec:quotient}
   \begin{equation}
     \pi^{-1}(\pi(U)) = \bigcup_{t\neq 0} tU = \bigcup_{t\neq 0}\{tp \mid p\in U\}.
   \end{equation}
-  Since multiplication by $t\neq 0$ is a homeomorphism of $\R_0^{n+1}$, the set $t U$ is open for any $t$, as is their union, $\RP^n$ is both Hausdorff and second-countable.
+  Since multiplication by $t\neq 0$ is a homeomorphism of $\R_0^{n+1}$, the set $t U$ is open for any $t$, as is their union, thus $\RP^n$ is both Hausdorff and second-countable.
 
   For each $i=0,\ldots,n$, define $\widetilde U_i := \{x\in\R^{n+1}_0 \mid x^i\neq0\}$, the set where the $i$-th coordinate is not $0$, and let $U_i = \pi(\widetilde U_i)\subset \RP^n$.
   Since $\widetilde U_i$ is open, $U_i$ is open.

aom.tex

@@ -235,7 +235,7 @@ \chapter*{Introduction}
   These theories are usually studied in the context of real numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
 \end{quotation}
 
-\newthought{In this sense}, our course will focus on generalizing the concepts of differentiation, integration and, up to some extent, differential equations on spaces that are more general than the standard Euclidean space.
+\newthought{In this sense}, our course will focus on generalizing the concepts of differentiation, integration and, up to some extent, differential equations to spaces that are more general than the standard Euclidean space.
 We will do this by trying to make everything look Euclidean and, in this sense, the Euclidean space $\R^n$ is going to be \emph{the} prototype of all manifolds: it won't just be our simplest example, we will see that locally every manifold looks like a Euclidean space.
 
 Euclidean spaces, and the Riemannian charts that you may have already encountered in the Geometry course, have a very strong property: they can be described with a set of \emph{global} coordinates.
@@ -264,17 +264,17 @@ \chapter*{Introduction}
 As complementery sources you can use the textbook~\cite{book:tu} or the extensive reference~\cite{book:lee}.
 You should have access to both books via the University library and, in addition, Lee's ebook can be downloaded via the University proxy on \href{https://link.springer.com/book/10.1007/978-1-4419-9982-5}{SpringerLink}.
 
-The book~\cite{book:McInerney} is a nice compact companion that develops most of the concepts of the course in the specific case of $\R^n$ and could provide further examples and food for thoughts.
-The books \cite{book:nicolaescu}\footnote{Beware of typos, there are many.}, \cite{book:crane} and \cite{lectures:nanda}, freely available from the authors' website, are not really suitable as references for this courses but provides fantastic resources for the readers that want to dig further and see where the material discussed in the course can lead.
+The book~\cite{book:McInerney} is a nice compact companion that develops most of the concepts of the course in the specific case of $\R^n$ and could provide further examples and food for thought.
+The books \cite{book:nicolaescu}\footnote{Beware of typos, there are many.}, \cite{book:crane} and \cite{lectures:nanda}, freely available from the authors' website, are not really suitable as references for this course but provide fantastic resources for the readers that want to dig further and see where the material discussed in the course can lead.
 Finally, a colleague mentioned~\cite{book:lang}. I don't have experience with this book but from a brief look it seems to follow a similar path as these lecture notes, so it might provide yet an alternative reference after all.
 
-The idea for the cut that I want to give to this course was inspired by the online \href{https://www.video.uni-erlangen.de/course/id/242}{Lectures on the Geometric Anatomy of Theoretical Physics} by Frederic Schuller, by the lecture notes of Stefan Teufel's Classical Mechanics course~\cite{lectures:teufel} (in German), by the classical mechanics book by Arnold~\cite{book:arnold} and by the Analysis of Manifold chapter in~\cite{book:thirring}.
+The idea for the cut that I want to give to this course was inspired by the online \href{https://www.video.uni-erlangen.de/course/id/242}{Lectures on the Geometric Anatomy of Theoretical Physics} by Frederic Schuller, by the lecture notes of Stefan Teufel's Classical Mechanics course~\cite{lectures:teufel} (in German), by the classical mechanics book by Arnold~\cite{book:arnold} and by the Analysis of Manifolds chapter in~\cite{book:thirring}.
 In some sense I would like this course to provide the introduction to geometric analysis that I wish was there when I prepared my \href{https://www.mseri.me/lecture-notes-hamiltonian-mechanics/}{first edition} of the Hamiltonian mechanics course.
 In addition to the reference above, these lecture notes have found deep inspiration from~\cite{lectures:merry,lectures:hitchin} (all freely downloadable from the respective authors' websites), and from the book~\cite{book:abrahammarsdenratiu}.
 
-I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his useful comments and corrections, and to Bram Brongers\footnote{You can also have a look at \href{https://fse.studenttheses.ub.rug.nl/25344/}{his bachelor thesis} to learn more about some interesting advanced topics in differential geometry.} for his comments, corrections and the appendices that he contributed to these notes.
+I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his invaluable suggestions, comments and corrections, and to Bram Brongers\footnote{You can also have a look at \href{https://fse.studenttheses.ub.rug.nl/25344/}{his bachelor thesis} to learn more about some interesting advanced topics in differential geometry.} for his comments, corrections and the appendices that he contributed to these notes.
 
-Many thanks also to: Wojtek Anyszka, Huub Bouwkamp, Daniel Cortlid, Anna de Bruijn, Luuk de Ridder, Mollie Jagoe Brown, Wietze Koops, Henrieke Krijgsheld, Levi Moes, Nicol\'as Moro, Magnus Petz, Lisanne Sibma, Bo Tielman, Jesse van der Zeijden, Jordan van Ekelenburg, Hanneke van Harten, Martin Daan van IJcken, Marit van Straaten, Dave Verweg and Federico Zadra for their comments and for reporting a number of misprints and corrections.
+Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Wojtek Anyszka, Huub Bouwkamp, Daniel Cortlid, Anna de Bruijn, Luuk de Ridder, Mollie Jagoe Brown, Wietze Koops, Henrieke Krijgsheld, Levi Moes, Nicol\'as Moro, Magnus Petz, Lisanne Sibma, Bo Tielman, Jesse van der Zeijden, Jordan van Ekelenburg, Hanneke van Harten, Martin Daan van IJcken, Marit van Straaten, Dave Verweg and Federico Zadra.
 
 \mainmatter