Small updates

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

2b-submanifolds.tex

@@ -54,15 +54,20 @@ \section{Inverse function theorem}
 Note that this theorem can fail for manifold with boundary.
 A counterexample\footnote{Exercise: why?} is given by the inclusion map $\cH^n \hookrightarrow \R^n$. 
 
-An important observation at this point is that the crucial property
-of the mapping in the inverse function theorem is the rank of its
-differential.
-If we think in euclidean terms and remember that the differential of a map
-is the best linear approximation to the map, then we can start
-wondering if we can use this as a tool to probe the structure of manifolds
-and their tangent spaces.
-
-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.
+An important observation at this point is that the rank of the mapping
+is a crucial property in the inverse function theorem and it is really a
+property of its differential.
+If we map our manifold via some function $F$ or via its charts into a
+euclidean space in such a way that the rank of the mapping remains fixed,
+and thus all the tangent spaces will be mapped to euclidean spaces of same dimensions,
+we may be able to use the shape of the mapping itself to describe the
+shape of the manifold.
+
+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 show that any manifold locally looks like a projection or an inclusion.
+The tool to get there is the following, we will see more clearly the link with
+projections and inclusions in the next subsection.
 
 \begin{theorem}[Rank theorem]\label{thm:rank}
   Let $F : M^m \to N^n$ be a smooth function between smooth manifolds without boundary\footnote{The theorem can be extended to allow $M$ with boundary and $N$ without boundary assuming $\ker dF_p \not\subseteq T_p\partial M$ but we will omit this case here to keep the discussion more contained and avoid unnecessary technicalities.}.

2c-vectorbdl.tex

@@ -166,7 +166,7 @@ \section{Vector bundles}
   Let $\pi:E \to M$ be a smooth vector bundle of rank $k$ over $M$. Let $U,V\subseteq M$, $U\cap V\neq \emptyset$.
   If $\varPhi : \pi^{-1}(U) \to U \times \R^k$ and $\Psi: \pi^{-1}(V) \to V \times \R^k$ are two smooth local trivializations of $E$, then there exists a smooth map $\tau: U\cap V \to GL(k, \R)$ such that
   \begin{align}
-    \varPhi\circ\Psi^{-1} : (U\cap V)\times\R^k &\to (U\cap V)\times \R^k \to (U\cap V)\times \R^k \\
+    \varPhi\circ\Psi^{-1} : (U\cap V)\times\R^k &\to \pi^{-1}(U\cap V) \to (U\cap V)\times \R^k \\
                             (p,v) &\mapsto (p, \tau(p) v).
   \end{align}
 \end{lemma}