Updates on vector and tensor bundles

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

2c-vectorbdl.tex

@@ -102,8 +102,8 @@ \section{Vector bundles}\label{sec:vec-bdls}
 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 $\R^n\times\R^k$.
+  Let $(U_\alpha, \varphi_\alpha)$ be a smooth structure on $M$ subordinate to the given open cover, see Theorem~\ref{thm:partitionof1}.
+  We need to use this, and the given maps from the statement, to define charts from some neighbourhood of fibers $E_p$ to $\R^n\times\R^k$.
   For each $p\in M$, choose an open neighbourhood $V_p \supseteq U_\alpha$ 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}
@@ -148,7 +148,7 @@ \section{Vector bundles}\label{sec:vec-bdls}
       \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.
+  In a similar way, the coordinate representation of $\pi : E \to M$ is $\varphi_\alpha \circ \pi \circ \widetilde{\varphi}_\alpha^{-1} (x,v) = x$, 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}.

4-cotangentbdl.tex

@@ -440,13 +440,13 @@ \section{One-forms and the cotangent bundle}
 The pullback is a rather pervasive concept, and does provide us a new way to explore vector bundles.
 
 \begin{example}[The pullback bundle]
-	Let $F:M\to N$ be a smooth map between manifolds. Suppose that $\pi: E \to N$ be a vector bundle of rank $r$ over $N$.
-	Then $M\times E$ is a trivial bundle over $M$ with constant fibre $E$.
+	Let $F: M\to N$ be a smooth map between manifolds. Suppose that $\pi: E \to N$ is a vector bundle of rank $r$ over $N$.
+	Then we can think of $M\times E$ as a trivial (fiber)\marginnote{A fiber bundle is a bundle where the fibers are not necessarily vector spaces, but can be in general topological spaces. It is a good exercise to try and modify the definition of vector bundles so that it applies to this case (hint: drop any direct or indirect appearance of linearity). We will not discuss them in this course and for the sake of this example we don't really need to know more about them. For more details you can refer to \cite[Chapter 10]{book:lee}.} bundle over $M$ with constant fibre $E$.
 	You may think that this is yet another trivial example, but it allows us to define the \emph{pullback bundle $F^* E$}: let
 	\begin{equation}
 		F^* E := \left\lbrace (p, v) \in M\times E \mid F(p) = \pi(v)\right\rbrace,
 	\end{equation}
-	with the projection $\Pi_1:F^* E \to M$.
+	with the projection $\Pi_1: F^* E \to M$.
 	The fibre of $F^*E$ over $p\in M$, then, is $\{p\}\times E_{F(p)}$, which under $\Pi_2:F^* E \to E$ is diffeomorphic to $E_{F(p)}$.
 	If $\varphi : \pi^{-1}(U) \to U\times\R^r$ is a bundle diffeomorphism for $E$, then $\varphi\circ\Pi_2: \Pi_1^{-1}(F^{-1}(U)) \to U\times\R^r$ is a bundle diffeomorphism for $F^*E$.
 	This $F^*E$ is a vector bundle of rank $r$ over $M$.

aom.tex

@@ -214,7 +214,7 @@
 	\setlength{\parskip}{\baselineskip}
 	Copyright \copyright\ \the\year\ \thanklessauthor
 
-	\par Version 1.6.5 -- \today
+	\par Version 1.6.6 -- \today
 
 	\vfill
 	\small{\doclicenseThis}