Clarify definition

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

2c-vectorbdl.tex

@@ -1,6 +1,6 @@
 What we have seen in the previous chapter is our first example of vector bundle, which is just a way to call a vector space depending continuously (or smoothly) on some parameters, for example points on a manifold.
 
-\section{Vector bundles}
+\section{Vector bundles}\label{sec:vec-bdls}
 
 \begin{definition}\label{def:vector_bundle}
   A \emph{vector bundle of rank $r$} over a manifold $M$ is a manifold $E$ together with a smooth surjective map $\pi : E \to M$ such that, for all $p\in M$, the following properties hold:
@@ -72,7 +72,7 @@ \section{Vector bundles}
 \end{definition}
 %
 \begin{marginfigure}
-    \includegraphics{images/2.10-subbundle.pdf}
+  \includegraphics{images/2.10-subbundle.pdf}
 \end{marginfigure}
 %
 
@@ -92,67 +92,67 @@ \section{Vector bundles}
     \item there exists an open cover $\{U_\alpha\}_{\alpha\in A}$ of $M$;
     \item for each $\alpha\in A$, there exists a bijection $\varPhi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha\times\R^k$ whose restriction $\varPhi_\alpha|_{E_p} : E_p \to \{p\}\times\R^k \sim \R^k$ is a linear isomorphism;
     \item for each $\alpha,\beta \in A$ with $U_{\alpha\beta}:=U_\alpha \cap U_\beta \neq \emptyset$, there exists a matrix-valued smooth map $\tau_{\alpha\beta}: U_{\alpha\beta} \to GL(k,\R)$ such that\footnote{Here $\tau_{\alpha\beta}(p) v$ denotes the usual product of the $k\times k$ matrix $\tau_{\alpha\beta}(p)$ with the vector $v\in\R^k$.}
-      \begin{align}
-        \varPhi_\alpha \circ \varPhi_\beta^{-1} : U_{\alpha\beta}\times\R^k &\to U_{\alpha\beta}\times\R^k\\
-                                                  (p, v) & \mapsto (p, \tau_{\alpha\beta}(p) v).
-      \end{align}
+          \begin{align}
+            \varPhi_\alpha \circ \varPhi_\beta^{-1} : U_{\alpha\beta}\times\R^k & \to U_{\alpha\beta}\times\R^k         \\
+            (p, v)                                                              & \mapsto (p, \tau_{\alpha\beta}(p) v).
+          \end{align}
   \end{enumerate}
   Then $E$ has a unique topology and a smooth structure making it into a smooth manifold without boundary and a smooth rank $k$ vector bundle over $M$ with $\pi$ as its projection and $\{(U_\alpha, \varPhi_\alpha)\}$ as smooth local trivializations.
 \end{theorem}
 The maps $\tau_{\alpha\beta}$ are called \emph{transition function}s between the local trivializations.
 \begin{proof}
   \newthought{Part 1. $E$ has a structure of smooth manifold}.
-Let $(U_\alpha, \varphi_\alpha)$ be a smooth structure on $M$ adapted to the given open cover.
-We need to use this, and the given maps from the statement, to define charts from $E_p$ to $\R^n\times\R^k$.
-For each $p\in M$, choose an open neighbourhood $V_p \subseteq U_\alpha$ for some $\alpha\in A$.
-Observe that $\pi^{-1}(V_p) \subseteq E_p$ and $\widetilde{V}_p := \varphi_\alpha(V_p) \subseteq \R^n$ and therefore it may be natural to consider the collection $\{(\pi^{-1}(V_p), \widetilde{\varphi}_p)\mid p\in M\}$, where
-\begin{equation}
-  \widetilde{\varphi}_p := (\varphi_\alpha\times \id_k) \circ \varPhi_\alpha : \pi^{-1}(V_p) \xrightarrow{\varPhi_\alpha} V_p\times\R^k \xrightarrow{\varphi_\alpha\times \id_k} \widetilde{V}_p\times \R^n,
-\end{equation}
-as a candidate to apply the Smooth Manifold Lemma~\ref{lem:manifold_chart}.
-This would give $E$ both a topology and a smooth structure.
+  Let $(U_\alpha, \varphi_\alpha)$ be a smooth structure on $M$ adapted to the given open cover.
+  We need to use this, and the given maps from the statement, to define charts from $E_p$ to $\R^n\times\R^k$.
+  For each $p\in M$, choose an open neighbourhood $V_p \subseteq U_\alpha$ for some $\alpha\in A$.
+  Observe that $\pi^{-1}(V_p) \subseteq E_p$ and $\widetilde{V}_p := \varphi_\alpha(V_p) \subseteq \R^n$ and therefore it may be natural to consider the collection $\{(\pi^{-1}(V_p), \widetilde{\varphi}_p)\mid p\in M\}$, where
+  \begin{equation}
+    \widetilde{\varphi}_p := (\varphi_\alpha\times \id_k) \circ \varPhi_\alpha : \pi^{-1}(V_p) \xrightarrow{\varPhi_\alpha} V_p\times\R^k \xrightarrow{\varphi_\alpha\times \id_k} \widetilde{V}_p\times \R^n,
+  \end{equation}
+  as a candidate to apply the Smooth Manifold Lemma~\ref{lem:manifold_chart}.
+  This would give $E$ both a topology and a smooth structure.
 
-Checking the requirements for the Smooth Manifold Lemma is then relatively straightforward.
-First of all, observe that $\widetilde{\varphi}_p$ is a bijection from $\pi^{-1}(V_p)$ onto $\widetilde{V}_p\times\R^k \subseteq \R^{n+k}$ since it is the composition of bijective maps.
-By construction, for $p,q\in M$, we have
-\begin{equation}
-  \widetilde{\varphi}_p(\pi^{-1}(V_p) \cap \pi^{-1}(V_q)) = \phi_p(V_p\cap V_q)\times \R^k
-\end{equation}
-which is open in $\R^{n+k}$ since $\varphi_p$ is a homeomorphism onto an open $V_p\subseteq\R^n$.
-Finally, on the overlap of two charts, there are $\alpha, \beta\in A$ such that
-\begin{equation}
-  \widetilde{\varphi}_p \circ \widetilde{\varphi}_q^{-1} =
-  (\varphi_\alpha \times \id_k) \circ \varPhi_\alpha \circ \varPhi_\beta^{-1} \circ (\varphi_\beta \times \id_k)^{-1}.
-\end{equation}
-Such transition map is a diffeomorphism since it is the composition of diffeomorphisms.
-Therefore properties (i) and (ii) of the theorem are satisfied.
+  Checking the requirements for the Smooth Manifold Lemma is then relatively straightforward.
+  First of all, observe that $\widetilde{\varphi}_p$ is a bijection from $\pi^{-1}(V_p)$ onto $\widetilde{V}_p\times\R^k \subseteq \R^{n+k}$ since it is the composition of bijective maps.
+  By construction, for $p,q\in M$, we have
+  \begin{equation}
+    \widetilde{\varphi}_p(\pi^{-1}(V_p) \cap \pi^{-1}(V_q)) = \phi_p(V_p\cap V_q)\times \R^k
+  \end{equation}
+  which is open in $\R^{n+k}$ since $\varphi_p$ is a homeomorphism onto an open $V_p\subseteq\R^n$.
+  Finally, on the overlap of two charts, there are $\alpha, \beta\in A$ such that
+  \begin{equation}
+    \widetilde{\varphi}_p \circ \widetilde{\varphi}_q^{-1} =
+    (\varphi_\alpha \times \id_k) \circ \varPhi_\alpha \circ \varPhi_\beta^{-1} \circ (\varphi_\beta \times \id_k)^{-1}.
+  \end{equation}
+  Such transition map is a diffeomorphism since it is the composition of diffeomorphisms.
+  Therefore properties (i) and (ii) of the theorem are satisfied.
 
-Property (iii) follows from the fact that $M$ is a manifold and thus the open cover $\{V_p | p \in M\}$ has a countable subcover.
+  Property (iii) follows from the fact that $M$ is a manifold and thus the open cover $\{V_p | p \in M\}$ has a countable subcover.
 
-We are left with the Hausdorff property (iv).
-This can be checked in the same vein as we did for the analogous proof in the case of the tangent bundle:
-if two vectors belong to the same space $E_p$, then they belong to one of the charts we have constructed and we can separate them taking the preimages of two disjoint open neighbourhoods of $\R^k$ containing them; if the two points belong to two distinct spaces $E_p$ and $E_q$, $p\neq q$, we can pick two disjoint neighbourhoods $V_p$ and $V_q$ so that their preimages $\pi^{-1}(V_p)$ and $\pi^{-1}(V_q)$ are disjoint coordinate neighbourhoods separating the two points.
+  We are left with the Hausdorff property (iv).
+  This can be checked in the same vein as we did for the analogous proof in the case of the tangent bundle:
+  if two vectors belong to the same space $E_p$, then they belong to one of the charts we have constructed and we can separate them taking the preimages of two disjoint open neighbourhoods of $\R^k$ containing them; if the two points belong to two distinct spaces $E_p$ and $E_q$, $p\neq q$, we can pick two disjoint neighbourhoods $V_p$ and $V_q$ so that their preimages $\pi^{-1}(V_p)$ and $\pi^{-1}(V_q)$ are disjoint coordinate neighbourhoods separating the two points.
 
-With this property also satisfied, Lemma~\ref{lem:manifold_chart} gives $E$ the structure of smooth manifold without boundary.
+  With this property also satisfied, Lemma~\ref{lem:manifold_chart} gives $E$ the structure of smooth manifold without boundary.
 
-\newthought{Part 2. The maps $\Phi_\alpha$ define a smooth local trivialization}.
-Again, by construction, the maps $\varPhi_\alpha : \pi^{-1}(V_p) \to V_p\times\R^k$ are diffeomorphisms.
-Indeed, their coordinate representations with respect to the charts introduced above is just the identity\footnote{Exercise: expand the definitions and show that the following diagram commutes}:
-\begin{equation}
-  \begin{tikzcd}
-    {\pi^{-1}(V_p)} & {V_p\times\R^k} \\
-    {\widetilde{V}_p\times\R^k} & {\widetilde{V}_p\times\R^k}
-    \arrow["{\varPhi_\alpha}", from=1-1, to=1-2]
-    \arrow["{\varphi_p\times\id_k}", from=1-2, to=2-2]
-    \arrow["{\widetilde{\varphi}_p}"', from=1-1, to=2-1]
-    \arrow["{\id_{n+k}}"', from=2-1, to=2-2]
-  \end{tikzcd}.
-\end{equation}
-In a similar way, the coordinate representation of $\pi : E \to M$ is $\varphi_\alpha \circ \pi \circ \widetilde{\varphi}_\alpha (x,v) = x$, so $\pi$ is smooth.
-Finally, $\Phi_\alpha$ satisfies all conditions to be a smooth local trivialization since $\varPhi_\alpha$ is linear by hypothesis and $\pi_1 \circ \varPhi_\alpha = \pi$, which follows from $\varPhi_\alpha(E_p) = \{p\}\times \R^k$.
+  \newthought{Part 2. The maps $\Phi_\alpha$ define a smooth local trivialization}.
+  Again, by construction, the maps $\varPhi_\alpha : \pi^{-1}(V_p) \to V_p\times\R^k$ are diffeomorphisms.
+  Indeed, their coordinate representations with respect to the charts introduced above is just the identity\footnote{Exercise: expand the definitions and show that the following diagram commutes}:
+  \begin{equation}
+    \begin{tikzcd}
+      {\pi^{-1}(V_p)} & {V_p\times\R^k} \\
+      {\widetilde{V}_p\times\R^k} & {\widetilde{V}_p\times\R^k}
+      \arrow["{\varPhi_\alpha}", from=1-1, to=1-2]
+      \arrow["{\varphi_p\times\id_k}", from=1-2, to=2-2]
+      \arrow["{\widetilde{\varphi}_p}"', from=1-1, to=2-1]
+      \arrow["{\id_{n+k}}"', from=2-1, to=2-2]
+    \end{tikzcd}.
+  \end{equation}
+  In a similar way, the coordinate representation of $\pi : E \to M$ is $\varphi_\alpha \circ \pi \circ \widetilde{\varphi}_\alpha (x,v) = x$, so $\pi$ is smooth.
+  Finally, $\Phi_\alpha$ satisfies all conditions to be a smooth local trivialization since $\varPhi_\alpha$ is linear by hypothesis and $\pi_1 \circ \varPhi_\alpha = \pi$, which follows from $\varPhi_\alpha(E_p) = \{p\}\times \R^k$.
 
-\newthought{Part 3. The smooth structure is unique}.
-Since the $\varPhi_\alpha$ have to be diffeomorphisms onto their images, any other atlas will have to contain the family of charts defined here, and therefore they must be in the same equivalence class, that is, in the same smooth structure.
+  \newthought{Part 3. The smooth structure is unique}.
+  Since the $\varPhi_\alpha$ have to be diffeomorphisms onto their images, any other atlas will have to contain the family of charts defined here, and therefore they must be in the same equivalence class, that is, in the same smooth structure.
 \end{proof}
 
 \begin{exercise}
@@ -166,8 +166,8 @@ \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 \pi^{-1}(U\cap V) \to (U\cap V)\times \R^k \\
-                            (p,v) &\mapsto (p, \tau(p) v).
+    \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}
 \begin{exercise}
@@ -200,8 +200,8 @@ \section{Vector bundles}
       \varPhi\circ\varPsi^{-1}(p, (v^1, v^2)) = (p, \tau(p)(v^1, v^2)), \quad
       \tau(p) :=
       \begin{pmatrix}
-        \tau^1(p) & 0 \\
-        0 & \tau^2(p)
+        \tau^1(p) & 0         \\
+        0         & \tau^2(p)
       \end{pmatrix}.
     \end{equation}
     Here $\tau^1$ and $\tau^2$ denote transition functions of $E^1$ and $E^2$ respectively.
@@ -232,7 +232,7 @@ \section{Sections of vector bundles}
 \end{remark}
 
 \begin{example}[Vector fields in $\R^n$]
-If $E = M\times \R^r$, $M\subset\R^n$, then for any smooth map $F: M\to\R^r$ we have a section $S\in\Gamma(E)$ defined by $S(p) = (p, F(p))$. This is a classical euclidean vector field: a map that associates vectors to points (see Figure~\ref{fig:vectorfield-rn}).
+  If $E = M\times \R^r$, $M\subset\R^n$, then for any smooth map $F: M\to\R^r$ we have a section $S\in\Gamma(E)$ defined by $S(p) = (p, F(p))$. This is a classical euclidean vector field: a map that associates vectors to points (see Figure~\ref{fig:vectorfield-rn}).
   Notice, in particular, that functions $f\in C^\infty(M)$ are sections of the trivial bundle $M\times\R$:
   \begin{equation}
     \Gamma(M\times\R) \simeq C^\infty(M).
@@ -242,9 +242,9 @@ \section{Sections of vector bundles}
 One can sometimes distinguish non--isomorphic bundles by looking at the complement of their zero section: since any vector bundle isomorphism $h:E_1\to E_2$ must map the zero section of $E_1$ onto the zero section of $E_2$, the complements of the zero sections in $E_1$ and $E_2$ must be homeomorphic.
 
 \begin{marginfigure}
-    \includegraphics{2_9-vfield.pdf}
-    \caption{A vector field ``attaches'' vectors to points.}%
-    \label{fig:vectorfield-rn}
+  \includegraphics{2_9-vfield.pdf}
+  \caption{A vector field ``attaches'' vectors to points.}%
+  \label{fig:vectorfield-rn}
 \end{marginfigure}
 
 Even though, as we have seen, \emph{locally} $TM$ is diffeomorphic to $M\times\R^n$, this is not true in general with one exception.
@@ -308,9 +308,8 @@ \section{Sections of vector bundles}
   \begin{enumerate}
     \item Show that the tangent bundle of $M$ in $N$, given by $T\widetilde M := dF(TM) \subset TN\big|_{\widetilde M}$, is a subbundle of $TN\big|_{\widetilde M}$ by providing explicit local trivializations in terms of the charts $(U, \varphi)$ for $M$.
     \item Assume that there exist a smooth function $\Phi:N\to\R^{n-m}$ such that $\widetilde M := \{p\in N \mid \Phi(p) = 0\}$ and $d\Phi_p$ has full rank for all $p\in\widetilde M$. Prove that\footnote{Here $T\,N|_{\widetilde{M}}$ denotes the tangent bundle of $N$ restricted to the base points in $\widetilde{M}$.}
-      \begin{equation}
-        T\widetilde{M} = \left\{(p,v)\in T\,N|_{\widetilde{M}} \mid v\in\ker(d\Phi_p)\right\}.
-      \end{equation}
+          \begin{equation}
+            T\widetilde{M} = \left\{(p,v)\in T\,N|_{\widetilde{M}} \mid v\in\ker(d\Phi_p)\right\}.
+          \end{equation}
   \end{enumerate}
 \end{exercise}
-