Fix some issues

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

2b-submanifolds.tex

@@ -1,5 +1,5 @@
 With differentials of smooth functions at hand, we are ready to discuss submanifolds: smaller manifolds sitting inside larger ones.
-We have already seen an example at the begininng of the course.
+We have already seen an example at the beginning of the course.
 In Exercise~\ref{exe:subsetsmanifolds}, we proved that any open subset $U\subseteq M$ can be made into a smooth manifold with a differentiable structure induced by the one of $M$.
 These, somehow trivial, submanifolds are called \emph{open submanifolds}.
 But there are many other examples beyond these ones.
@@ -65,7 +65,7 @@ \section{Inverse function theorem}
 In fact, if we restrict our attention to constant rank maps, that is, maps whose rank is the same at all points on the manifold, we can go quite a long way and the tool to get there is the following.
 
 \begin{theorem}[Rank theorem]\label{thm:rank}
-  Let $F : M^m \to N^n$ be a smooth function between smooth manifolds.
+  Let $F : M^m \to N^n$ be a smooth function between smooth manifolds without boundary\footnote{The theorem can be extended to manifolds with boundary 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 $F$ has a coordinate representation of the form
   \begin{align}
@@ -277,12 +277,12 @@ \section{Embeddings, submersions and immersions}
   \caption{Theorem~\ref{thm:rank}, case of Proposition~\ref{prop:local_embedding}, in a picture.}
 \end{marginfigure}
 \begin{proposition}\label{prop:local_embedding}
-  Let $M^m$ and $N^n$ be smooth manifolds and $F:M\to N$ an immersion.
-  Then for any $p\in M$, there exists a neighbourhood $U$ of $p$ such that $F\big|_U$ is an embedding onto its image.
+  Let $M^m$ and $N^n$ be smooth manifolds without boundary\footnote{This is not necessary: the result holds also on manifolds with boundary but we need a modified version of the Rank Theorem in that case.} and $F:M\to N$ a smooth function.
+  Then $F$ is an immersion if and only if $F$ is a \emph{local embedding}, that is, for any $p\in M$, there exists a neighbourhood $U$ of $p$ such that $F\big|_U : U \to N$ is an embedding.
 \end{proposition}
 \begin{exercise}
   Prove Proposition~\ref{prop:local_embedding}. \\
-  \textit{\small Hint: use the Rank Theorem~\ref{thm:rank} and construct appropriate charts}
+  \textit{\small Hint: for the nontrivial direction use the Rank Theorem~\ref{thm:rank} and construct appropriate charts}
 \end{exercise}
 
 In fact we can say more if the manifold is compact.
@@ -392,7 +392,7 @@ \section{Embeddings, submersions and immersions}
 
 \begin{example}\label{ex:s2}
   The sphere $\bS^2 = \{x\in\R^3 \mid \|x\| = 1\}$ is a $2$-dimensional submanifold of $N=\R^3$.
-  This is an immediate consquence of Theorem~\ref{thm:impl_fun}: let $\psi(x) = \|x\|^2 -1 : \R^3 \to \R$, then $\psi$ is smooth, $\bS^2 = \{x\in\R^3\mid\psi(x)=0\}$ and, denoting $t$ the coorindate on $\R$, $d\psi_x(v)= v^i \frac{\partial \psi}{\partial x^i}|_x \frac{\partial}{\partial t}|_0 = (2x\cdot v) \frac{\partial}{\partial t}|_0$, that is, as a 1x3 matrix $d\psi_x = 2(x^1\; x^2\; x^3)$ so it is of maximal rank $1$ for all $x\in\bS^2$.
+  This is an immediate consequence of Theorem~\ref{thm:impl_fun}: let $\psi(x) = \|x\|^2 -1 : \R^3 \to \R$, then $\psi$ is smooth, $\bS^2 = \{x\in\R^3\mid\psi(x)=0\}$ and, denoting $t$ the coorindate on $\R$, $d\psi_x(v)= v^i \frac{\partial \psi}{\partial x^i}|_x \frac{\partial}{\partial t}|_0 = (2x\cdot v) \frac{\partial}{\partial t}|_0$, that is, as a 1x3 matrix $d\psi_x = 2(x^1\; x^2\; x^3)$ so it is of maximal rank $1$ for all $x\in\bS^2$.
 \end{example}
 
 \begin{example}

aom.tex

@@ -213,7 +213,7 @@
   \setlength{\parskip}{\baselineskip}
   Copyright \copyright\ \the\year\ \thanklessauthor
 
-  \par Version 1.4 -- \today
+  \par Version 1.4.1 -- \today
 
   \vfill
   \small{\doclicenseThis}