From 41fd02279f40b484a2353ff680ed2b54939bd876 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Wed, 1 Nov 2023 00:43:24 +0100 Subject: [PATCH] Add exercise on pullback bundles Signed-off-by: Marcello Seri --- 2c-vectorbdl.tex | 2 +- 4-cotangentbdl.tex | 5 ++++- 2 files changed, 5 insertions(+), 2 deletions(-) diff --git a/2c-vectorbdl.tex b/2c-vectorbdl.tex index 9fb5277..99a6dd1 100644 --- a/2c-vectorbdl.tex +++ b/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} diff --git a/4-cotangentbdl.tex b/4-cotangentbdl.tex index c0d367a..bb948b5 100644 --- a/4-cotangentbdl.tex +++ b/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}