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Signed-off-by: Marcello Seri <marcello.seri@gmail.com>
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mseri committed Jan 2, 2021
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Expand Up @@ -454,9 +454,9 @@ \section{Tensor bundles}
\end{example}

\begin{exercise}[\textit{[homework 3]}]
Let $F:N\to P$ and $G:M\to N$ two diffeomorphisms of smooth manifolds.
Let $F:M\to N$ and $G:N\to P$ two diffeomorphisms of smooth manifolds.
\begin{enumerate}
\item Show that the chain rule $(F\circ G)_* = F_* \circ G_*$ holds.
\item Show that the chain rule $(G\circ F)_* = G_* \circ F_*$ holds.
\item Show that our previous definition\footnote{That is, Definition~\ref{def:pullback0s} -- which includes the pullback from Definition~\ref{def:pullback1f}.} of pullback is a particular case of the following general definition of a \emph{pullback of $(r,s)$-tensor fields by $F$}:
\begin{equation}
F^* := (F^{-1})_* : \cT_s^r(N) \to \cT_s^r(M).
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