Exercise 2.11

Question

extbf{Proposition 2.10.} Let $M, N$, and $P$ be smooth manifolds with or without boundary.
\begin{enumerate}[label=(\alph*)]
\item Every constant map c: $M ightarrow N$ is smooth.
\item The identity map of $M$ is smooth.
\item If $U \subseteq M$ is an open submanifold with or without boundary, then the inclusion map $U \hookrightarrow M$ is smooth.
\item If $F: M ightarrow N$ and $G: N ightarrow P$ are smooth, then so is $G \circ F: M ightarrow P$.
\end{enumerate}

extbf{Exercise 2.11} Prove parts (a)–(c) of the preceding proposition.

Elu Thingol · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item Let $c:M 	o N$ be a constant map of value $c \in N$\footnote{We use $c$ to denote both the function $c:M 	o N$ and the constant value $c \in N$.}. Fix $p \in M$ and let $(U,\varphi)$ be a chart containing $p$. Picking any chart $(V,\psi)$ that contains the constant value $c \in N$ will do the job since $U \cap c^{-1}(V) = U \cap \mathbb{R} = U$ is open. The coordinate representation
    $$\psi \circ c \circ \varphi^{-1} = \psi(c) : \varphi(U) 	o \psi(V)$$
    is a constant function of value $\varphi(c)$. Since a constant function between subsets of Euclidean spaces is smooth in the sense of ordinary calculus, the map $c$ must be smooth.

\item Let $\operatorname{Id}:M 	o M$ be the identity map. Fix $p \in M$ and let $(U,\varphi)$ be a chart containing $p$. Then $(U,\varphi)$ is also a chart that contains $\operatorname{Id}(p)$ and satisfies $U \subseteq \operatorname{Id}(U) = U$. The coordinate representation
    $$\varphi \circ \operatorname{Id} \circ \varphi^{-1} = \operatorname{Id}_{\mathbb{R}^n} : \varphi(U) 	o \varphi(U)$$
    is an identity function. Since an identity function between subsets of Euclidean spaces is smooth in the sense of ordinary calculus, the map $\operatorname{Id}$ must be smooth.

\item Let $\iota: U 	o M$ be the inclusion map of $U$\footnote{I.e., the restriction of identity to $U$.}. Fix $p \in U$ and let $(\widetilde{U},\widetilde{\varphi})$ be a chart of $U$ containing $p$. Note that $(U,\varphi)$ is also a chart of $M$ that in particular contains $\iota(p)=p$ and satisfies $\iota(U) = \operatorname{Id}|_U(U) = U$. The coordinate representation is
    $$\varphi \circ \iota \circ \varphi^{-1} = \varphi \circ \operatorname{Id}|_U \circ \varphi^{-1} = \operatorname{Id}_{\mathbb{R}^n} : \varphi(U) 	o \varphi(U).$$
    By the same argument as in the previous part of the exercise, $\iota$ is smooth.
\end{enumerate}

Exercise 2.11

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Exercise 2.11

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