Exercise 3.7 (Properties of differentials)

Question

extbf{Proposition 3.6 (Properties of Differentials).} Let $M, N$, and $P$ be smooth manifolds with or without boundary, let $F: M ightarrow N$ and $G: N ightarrow P$ be smooth maps, and let $p \in M$.
\begin{enumerate}[label=(\alph*)]
\item $d F_p: T_p M ightarrow T_{F(p)} N$ is linear.
\item $d(G \circ F)_p=d G_{F(p)} \circ d F_p: T_p M ightarrow T_{G \circ F(p)} P$.
\item $d\left(\operatorname{Id}_Might)_p=\operatorname{Id}_{T_p} M: T_p M ightarrow T_p M$.
\item If $F$ is a diffeomorphism, then $d F_p: T_p M ightarrow T_{F(p)} N$ is an isomorphism, and $\left(d F_pight)^{-1}=d\left(F^{-1}ight)_{F(p)}$.
\end{enumerate}

extbf{Exercise 3.7.} Prove Proposition 3.6.

Elu Thingol · Accepted Answer

\newcommand{\diff}{\mathrm{d}}
\newcommand{\Id}{\operatorname{Id}}
\begin{enumerate}[label=(\alph*)]
    \item Let $v,w \in T_pM$ and $c \in \mathbb{R}$ be arbitrary. For all $f \in C^{\infty}(N)$, we have:
    \begin{align*}
       \diff F_p (c\cdot v + w) (f) &= (c \cdot v + w) (f \circ F)\
       &= c\cdot v(f \circ F) + w(f \circ F)\
       &= c \cdot \diff F_p (v)(f) + \diff F_p(w)(f).
    \end{align*}
    Since we have verified the function equality for all inputs $f$, it follows that
    \begin{equation*}
       \diff F_p (cv + w) = c \cdot \diff F_p (v) + \diff F_p(w).
    \end{equation*}

\item  Let $v\in T_pM$ be arbitrary. For all $f \in C^{\infty}(P)$, we have:
    \begin{equation*} \def\abstand{30pt}
    \begin{array}{llr}
        \textcolor{blue}{\underbrace{\diff G \circ \diff F (v)}_{u \in T_{G(F(p))}P}} \textcolor{gray}{\underbrace{\left( f \right)}_{\in C^{\infty}(P)}} 
        %\stackrel{\text{def}}{=}
        &=&
        \diff G ( \underbrace{\textcolor{blue}{\diff F(v)}}_{\textcolor{blue}{w \in T_{F(p)}N}} ) \textcolor{gray}{\underbrace{\left( f \right)}_{\in C^{\infty}(P)}}\[\abstand]
        %\stackrel{\text{def}}{=} 
        &=&
        \underbrace{\textcolor{blue}{\diff F(v)}}_{\textcolor{blue}{w \in T_{F(p)}N}}  \textcolor{gray}{\underbrace{\left( f \circ G \right)}_{\in C^{\infty}(N)}}\[\abstand]
        %\stackrel{\text{def}}{=} 
        &=&
        \textcolor{blue}{v}  \textcolor{gray}{\underbrace{\left( f \circ G \circ F \right)}_{\in C^{\infty}(M)}}\[\abstand]
        %\stackrel{\text{def}}{=}
        &=&
        \textcolor{blue}{\underbrace{\diff (G \circ F) (v)}_{u \in T_{G(F(p))}P}} \textcolor{gray}{\underbrace{\left( f \right)}_{\in C^{\infty}(P)}} 
    \end{array}
    \end{equation*}
    Since we have verified the function equality for all inputs $f$, it follows that
    \begin{equation*}
        d(G \circ F)_p (v) =(d G_{F(p)} \circ d F_p) (v).
    \end{equation*}

\item For all $f \in C^{\infty}(P)$, we have:
    \begin{align*}
        \diff(\Id)_p(v)(f) = v(f \circ \Id_M) = v(f) = \left(\Id_{T_pM}(v)\right)(f).
    \end{align*}
    Since we have verified the function equality for all inputs $f$, it follows that
    \begin{equation*}
        \diff(\Id_{M})_p(v) =  \Id_{T_pM}(v).
    \end{equation*}

\item Suppose that $F$ is a diffeomorphism. Since $\Id_M = F^{-1} \circ F$, by parts (b) and (c) of this exercise, we have
    \begin{align*}
        \Id_{T_p M} = \diff \left(\Id_{M}\right)_{p} = \diff (F^{-1} \circ F)_p = \diff \left(F^{-1}\right)_{F(p)} \circ \diff F_{p}.
    \end{align*}
    Similarly, since $\Id_N = F \circ F^{-1}$, we have
    \begin{align*}
        \Id_{T_p N} = \diff \left(\Id_{N}\right)_{F(p)} = \diff (F \circ F^{-1})_p = \diff F_{p} \circ \diff \left(F^{-1}\right)_{F(p)}.
    \end{align*}
    Hence, $\diff F_p$ and $\diff \left(F^{-1}\right)_{F(p)}$ are linear isomorphisms and inverses of each other.
\end{enumerate}

Exercise 3.7 (Properties of differentials)

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Exercise 3.7 (Properties of differentials)

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