Add exercise on pullback bundles

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

2c-vectorbdl.tex

@@ -180,7 +180,7 @@
 In the special case of two bundles $E^1 \to M$ and $E^2 \to M$ over the same manifold, this would lead to $E^1 \times E^2 \to M \times M$.
 It is natural to ask oneself if we can make a construction that combines multiple vector bundles over the same base space to a new bundle over that same base space.
 This is call the Whitney sum of the bundles.
-\begin{exercise}[Whitney sum]
+\begin{exercise}[Whitney sum]\label{ex:whitney}
   Let $\pi^1 : E^1 \to M$ and $\pi^2 : E^2 \to M$ be two smooth vector bundles over $M$ of rank $k^1$ and $k^2$ respectively.
   The Whitney sum of $E^1 \oplus E^2$ of $E^1$ and $E^2$ is the smooth vector bundles $\pi: E^1 \oplus E^2 \to M$ whose fibers
   \begin{equation}

4-cotangentbdl.tex

@@ -450,7 +450,10 @@ \section{One-forms and the cotangent bundle}
 \end{example}
 
 \begin{exercise}
-\textcolor{red}{TODO: add exercise on Whitney sum as pullback of diagonal $\Delta : X \to X\times X$}
+  Prove that the Whitney Sum\footnote{See Exercise~\ref{ex:whitney}.} 
+  of two vector bundles $\pi_1 : E_1 \to M$ and $\pi_2 : E_2 \to M$
+  is the pullback $\Delta^*(E_1 \times E_1)$ of their product bundle by the diagonal map
+  $\Delta : M \to M \times M$, $\Delta(x) = (x, x)$.
 \end{exercise}
 
 \section{Line integrals}