Fix typos and clarify text

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

Author: mseri

1-manifolds.tex

@@ -243,12 +243,12 @@ \section{Differentiable manifolds}
 
 With these at hand, let's jump into the definition of smooth manifolds.
 
-\begin{definition}\label{def:cratlas}\idxdef{Atlas}
+\begin{definition}\label{def:cratlas}\idxdef{Atlas}\idxdef{Open cover}
 	A \emph{smooth atlas} is a collection
 	\begin{equation}
 		\cA = \{\varphi_\alpha: U_\alpha \to V_\alpha \;\mid\; \alpha\in A\}
 	\end{equation}
-	of pairwise compatible charts that cover\footnote{I.e. such that $M = \cup_{\alpha\in A} U_\alpha$. One calls the set $\{U_\alpha \;\mid\; \alpha\in A\}$, covering $M$ with open sets, an \emph{open cover} of $M$. Here $A$ is some index set, not necessarily countable.} $M$.
+	of pairwise compatible charts that cover\footnote{I.e. such that $M = \cup_{\alpha\in A} U_\alpha$. One calls the set $\{U_\alpha \;\mid\; \alpha\in A\}$, covering $M$ with open sets, an \emph{open cover} of $M$. Here $A$ is some \emph{index set}, not necessarily countable.} $M$.
 
 	Two smooth atlases are \emph{equivalent} if their union is also a smooth atlas. That is if any two charts in the atlases are compatible.
 \end{definition}
@@ -270,7 +270,7 @@ \section{Differentiable manifolds}
 	\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}
 
-\begin{notation}
+\begin{notation}\idxdef{Smooth chart}\idxdef{Smooth coordinate map}
 	By a \emph{smooth chart $(U, \varphi)$} in a smooth manifold $M$ we mean a chart in the maximal atlas of the differentiable structure of $M$, in this case we call $\phi$ a \emph{smooth coordinate map} on $M$. We say that the chart is around $p$ or about $p$ if $p\in U$.
 \end{notation}
 
@@ -310,7 +310,7 @@ \section{Differentiable manifolds}
 \end{exercise}
 
 \vspace{1em}
-\begin{notation}\label{ntn:coords}
+\begin{notation}\label{ntn:coords}\idxdef{Local coordinates}\idxdef{Coordinate functions}\idxdef{Standard coordinates $r^i$}
 	We will stick to the notation of~\cite{book:tu}.
 	In the context of manifolds, denote $r^i: \R^n\to\R$, $1\leq i\leq n$, the standard coordinates on $\R^n$.
 	With this notation, if $e_i$ denotes the $i$th standard basis vector\footnote{Identified with the \emph{point} $(0,\ldots,0,\LaTeXunderbrace{1}_{i\mbox{th component}},0,\ldots,0) \in\R^n$.} in $\R^n$, then $r^i(e_j) = \delta^i_j$.
@@ -495,7 +495,7 @@ \section{Quotient manifolds}\label{sec:quotient}
 	\newthought{Let's first show that $\RP^n$ is a topological $n$-manifold}.
 	The structure of topological manifold follows directly from the Theorem~\ref{thm:openproj} and $\pi$ being open, so let's prove that.
 
-	\begin{exercise}
+	\begin{exercise}\label{exe:RPnmanifold}
 		Assume that $\pi$ is an open map. Show that $\RP^n$ is a Hausdorff and second-countable topological manifold of dimension $n$.
 	\end{exercise}
 
@@ -504,7 +504,7 @@ \section{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, thus $\RP^n$ 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 it follows from Exercise~\ref{exe:RPnmanifold} that $\RP^n$ 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.
@@ -515,7 +515,7 @@ \section{Quotient manifolds}\label{sec:quotient}
 	\end{align}
 	e.g. $\varphi_0([x^0, \ldots, x^n]) = (x^1/x_0, \ldots, x^n/x_0)$.
 	This map is well--defined because its value is unchanged by multiplying $x$ by a non-zero constant.
-	Moreover, $\varphi_i$ is continuous and invertible, the inverses can be computed explicitly\footnote{Check that the composition $\varphi_i \circ \varphi_i^{-1} = \id$. What happens if you define $\widetilde\varphi^{-1}_i(y^1, \ldots, y^n) := \left[y^1, \ldots, y^{i-1}, 42, y^{i+1}, \ldots, y^n\right]$, what would the corresponding $\widetilde\varphi_i$ be?} as
+	Moreover, $\varphi_i$ is continuous and invertible, the inverses can be computed explicitly\footnote{Check that the composition $\varphi_i \circ \varphi_i^{-1} = \id$. What happens if you define $\widetilde\varphi^{-1}_i(y^1, \ldots, y^n) := \left[y^1, \ldots, y^{i}, 42, y^{i+1}, \ldots, y^n\right]$, what would the corresponding $\widetilde\varphi_i$ be?} as
 	\begin{equation}
 		\varphi_i^{-1}(y^1,\ldots,y^n) = \left[y^1, \ldots, y^{i}, 1, y^{i+1}, \ldots, y^n\right].
 	\end{equation}
@@ -688,7 +688,7 @@ \section{Smooth maps and differentiability}\label{sec:smoothfn}
 The following corollary is just a restatement of Proposition~\ref{prop:smoothlocal}, but provides a useful perspective on the construction of smooth maps.
 
 \begin{proposition}[Gluing lemma for smooth maps]\idxthm{Gluing lemma for smooth maps}
-	Let $M$ and $N$ be two smooth manifolds and let $\{U_\alpha\mid\alpha\in A\}$ be an open cover of $M$.
+	Let $M$ and $N$ be two smooth manifolds and let $\{U_\alpha\mid\alpha\in A\}$ be an open cover of $M$ with some index set $A$.
 	Suppose that for each $\alpha\in A$ we are given a smooth map $F_
 		\alpha:U_\alpha\to N$ such that the maps agree on the overlaps: $F_\alpha|_{U_\alpha\cap U_\beta} = F_\beta|_{U_\alpha\cap U_\beta}$ for all $\alpha,\beta\in A$.
 	Then there exists a unique smooth map $F:M\to N$ such that $F|_{U_\alpha} = F_\alpha$ for each $\alpha\in A$.

aom.tex

@@ -287,7 +287,7 @@ \chapter*{Introduction}
 
 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.\medskip
 
-Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Jamara Admiraal, Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Anna de Bruijn, Daniel Cortlid, Harry Crane, Fionn Donogue, Jordan van Ekelenburg, Brian Elsinga, Hanneke van Harten, Martin Daan van IJcken, Mollie Jagoe Brown, Remko de Jong, Aron Karakai, Hanna Karwowska, Wietze Koops, Henrieke Krijgsheld, Justin Lin, Valeriy Malikov, Mar\'ia Diaz Marrero, Aiva Misieviciute, Levi Moes, Nicol\'as Moro, Alexandru Oprea, Magnus Petz, Jorian Pruim, Luuk de Ridder, Lisanne Sibma, Marit van Straaten, Bo Tielman, Dave Verweg, Ashwin Vishwakarma, Lars Wieringa, Federico Zadra, Tijmen van der Ree, and Jesse van der Zeijden.
+Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Jamara Admiraal, Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Anna de Bruijn, Daniel Cortlid, Harry Crane, Fionn Donogue, Jordan van Ekelenburg, Brian Elsinga, Hanneke van Harten, Martin Daan van IJcken, Mollie Jagoe Brown, Remko de Jong, Aron Karakai, Hanna Karwowska, Wietze Koops, Henrieke Krijgsheld, Justin Lin, Valeriy Malikov, Mar\'ia Diaz Marrero, Aiva Misieviciute, Levi Moes, Nicol\'as Moro, Alexandru Oprea, Magnus Petz, Jorian Pruim, Luuk de Ridder, Lisanne Sibma, Marit van Straaten, Bo Tielman, Dave Verweg, Ashwin Vishwakarma, Lisa Werkman, Lars Wieringa, Federico Zadra, Tijmen van der Ree, and Jesse van der Zeijden.
 
 \mainmatter