Further clarifications

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

1-manifolds.tex

@@ -164,7 +164,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
 \end{exercise}
 
 \begin{example}\label{ex:uball}
-	The \emph{closed} unit ball $D_1(0)$, where similarly as before
+	The \emph{closed} unit ball $D_1^n(0)$, where
 	\begin{equation}
 		D_r(x) := \{z\in\R^n \;\mid\; d(z,x) \leq r\},
 	\end{equation}
@@ -265,17 +265,15 @@ \section{Differentiable manifolds}
 	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{notation}
-	By a \emph{chart $(U, \varphi)$ about $p$} in a manifold $M$ we mean a chart in the differentiable structure of $M$ such that $p\in U$.
-\end{notation}
-
 \begin{definition}\label{def:diffmanifold}
 	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.}
-
-	If $(U, \phi)$ is a chart in the differentiable structure $\cA$ of $M$, we say that $(U, \phi$ is a \emph{smooth chart} on $M$ and that $\phi$ is a \emph{smooth coordinate map} on $M$.
 \end{definition}
 
+\begin{notation}
+	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}
+
 In colloquial language, a differentiable manifold is just a space covered by charts with differentiable transition maps.
 Note that not all topological manifolds can be made into smooth manifolds, but counterexamples are hard to construct and you need at least to go to dimension 4 (more on this on Remark~\ref{remark:smoothstr}).
 
@@ -915,8 +913,8 @@ \section{Partitions of unity}\label{sec:partition_of_unity}
 
 \section{Manifolds with boundary}\label{sec:mbnd}
 
-\newthought{The definition of manifolds has a serious limitation}, even though it is perfectly good to describe curves\footnote{E.g. the circle seen in Example~\ref{ex:S1emb}.} and surfaces\footnote{E.g. the $2$-spheres $\bS^2$.}, it fails to describe many natural objects like a \emph{closed} interval $[a,b]\in\R$ or the \emph{closed} disk $D_1(0)$ of Example~\ref{ex:uball}.
-Note that in each of these cases, both the interior and the boundary are smooth manifolds and their dimension differ by one\footnote{In the first case the interior $(a,b)$ is a $1$-manifold and the boundary, the set $\partial[a,b] = \{a,b\}$, is a $0$-manifold. In the second case the interior of $D_1(0)$ is the open unit ball, a $2$-manifold, and the boundary $\partial D_1(0)$ is the $1$-manifold $\bS^1$.}.
+\newthought{The definition of manifolds has a serious limitation}, even though it is perfectly good to describe curves\footnote{E.g. the circle seen in Example~\ref{ex:S1emb}.} and surfaces\footnote{E.g. the $2$-spheres $\bS^2$.}, it fails to describe many natural objects like a \emph{closed} interval $[a,b]\in\R$ or the \emph{closed} disk $D_1^2(0)$ of Example~\ref{ex:uball}.
+Note that in each of these cases, both the interior and the boundary are smooth manifolds and their dimension differ by one\footnote{In the first case the interior $(a,b)$ is a $1$-manifold and the boundary, the set $\partial[a,b] = \{a,b\}$, is a $0$-manifold. In the second case the interior of $D_1^2(0)$ is the open unit ball, a $2$-manifold, and the boundary $\partial D_1^2(0)$ is the $1$-manifold $\bS^1$.}.
 
 Let's do a step back and think about topological manifolds: since both the closed interval and the closed disk are closed sets, we have problems to make them locally euclidean in neighbourhoods of their boundaries.
 Can we modify our local model to resemble something with a boundary?
@@ -1131,7 +1129,7 @@ \section{Manifolds with boundary}\label{sec:mbnd}
 \end{exercise}
 
 \begin{exercise}
-	Let $M = D_1\subset \R^n$ be the $n$-dimensional closed unit ball from Example~\ref{ex:uball}.
+	Let $M = D_1^n = \{ x\in\R^n \mid \|x\|\leq 1\} \subset \R^n$ be the $n$-dimensional closed unit ball from Example~\ref{ex:uball}.
 	\begin{enumerate}
 		\item Show that $M$ is a topological manifold with boundary in which each point of $\partial M = \bS^{n-1}$ is a boundary point and each point in $\mathring M = \{x\in\R^n\mid\|x\|<1\}$ is an interior point.
 		\item Give a smooth structure to $M$ such that every smooth interior chart is a smooth chart for the standard smooth structure on $\mathring M$.