---- > [!definition] Definition. ([[product manifold]]) > Let $M_{1}$ and $M_{2}$ be [[smooth manifold|smooth manifolds]] of dimension $m_{1}$ and $m_{2}$ respectively. The [[product topology|product space]] $M_{1} \times M_{2}$ is naturally a [[smooth manifold]] of dimension $m_{1}+m_{2}$, called the **product manifold of $M_{1}$ and $M_{2}$.** > > Moreover, the projections $\pi_{1}:M_{1} \times M_{2} \to M_{1}$ and $\pi_{2}:M_{1} \times M_{2} \to M_{2}$ are [[smooth maps between manifolds|smooth]] and fit into a characteristic property: if $N$ is another [[smooth manifold]], then a map $f:N \to M_{1} \times M_{2}$ is [[smooth maps between manifolds|smooth]] if and only if $p_{i} \circ f$ is smooth for $i=1,2$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQA5EAX1PU1z5CKchWp0mrdgFkA+uQAEAHSV4AtvAVyATDz4gM2PASLbSxcQxZtEIOeT38jQ06W2XJNu7N3dxMKABzeCJQADMAJwg1JFEQHAgkMwlrdhU0LHlHEEjo2JoEpABmGispW3TM3RpGegAjGEYABQFjYRAIrECACxxs3JjEMnjExDiyrzD+qMHhwsRkhrAoYuJecJmkOdGSkCWVxABaIrXKbiA > \begin{tikzcd} > & & M_1 \\ > N \arrow[r, "f"] \arrow[rru, bend left] \arrow[rrd, bend right] & M_1 \times M_2 \arrow[ru, "\pi_1"] \arrow[rd, "\pi_2"'] & \\ > & & M_2 > \end{tikzcd} > \end{document} > ``` > > [!basicproperties] > > - For all $(p,q) \in M_{1} \times M_{2}$ there is a [[natural transformation|canonical isomorphism]] $T_{(p,q)} \cong T_{p} M_{1} \oplus T_{q}M_{2}$. The coordinate projections $\pi_{1}$, $\pi_{2}$ have [[differential of a smooth map between smooth manifolds|differentials]] equaling the [[linear projector|linear projectors]] onto each factor. > > WLOG focus on the projection $\pi_{1}:M_{1} \times M_{2} \to M_{1}$. > Let $\varphi \times \psi$ be a product chart on $M_{1} \times M_{2}$, say, $\varphi=(x^{1},\dots,x^{m_{1}}): (U \subset M_{1}) \to (\widehat{U} \subset \mathbb{R}^{m_{1}})$ and $\psi=(y^{1},\dots,y^{m_{2}}): (V \subset M_{2}) \to (\widehat{V} \subset \mathbb{R}^{m_{2}})$. Then $d\pi_{1} |_{(p,q)}$ defined to have [[Jacobian]] equaling the [[derivative]] of $\widehat{U} \times \widehat{V} \xrightarrow{(\varphi \times \psi)^{-1}} U \times V \xrightarrow{\pi_{1}}U \xrightarrow{\varphi} \widehat{U}$ > i.e., $Df$ where $f$ is given by $f(x,y)= \varphi \circ \pi_{1} \circ \big( \varphi ^{-1}(x), \psi ^{-1}(y) \big)=\varphi ( \varphi ^{-1} (x))=x.$ > This is the matrix $\begin{bmatrix} > \boldsymbol I_{m_{1} \times m_{1}} & \boldsymbol 0_{m_{1} \times m_{2}} > \end{bmatrix}$ > which is precisely the linear projection in question. > [!justification] > The [[Hausdorff space|Hausdorff]] and [[second-countable space|second-countability]] conditions follow from [[product of Hausdorff spaces is Hausdorff]] and [[second-countable space|subspaces preserve second-countability]]. If $M_{1}$ and $M_{2}$ have atlases $(U_{\alpha}, \varphi_{\alpha})_{\alpha \in I}$, $(U_{\beta}, \varphi_{\beta})_{\beta \in J}$ indexed by $I,J$ respectively, then the claim is that $M_{1} \times M_{2}$ has as [[atlas of a manifold|atlas]] $\mathscr{{A}}:=(U_{\alpha} \times U_{\beta}, \varphi_{\alpha} \times \varphi_{\beta})_{\alpha \in I, \beta \in J}.$ Indeed, it is clear that each $\varphi_{\alpha} \times \varphi_{\beta}$ is a [[homeomorphism]] onto its image (take as inverse the map $\varphi_{\alpha}^{-1} \times \varphi_{\beta}^{-1}$) and for any $\alpha', \beta' \in I, J$ we see that the map $\begin{align} (\varphi_{\alpha} \times \varphi_{\beta}) \circ (\varphi_{\alpha'} \times \varphi_{\beta'})^{-1} &=(\varphi_{\alpha} \times \varphi_{\beta}) \circ (\varphi_{\alpha'}^{-1} \times \varphi_{\beta'}^{-1}) \\ & = (\varphi_{\alpha} \circ \varphi_{\alpha'} ^{-1}) \times (\varphi_{\beta} \circ \varphi_{\beta'} ^{-1} ) \end{align}$ is [[continuously differentiable|smooth]] on Euclidean space because the factors $(\varphi_{\alpha} \circ \varphi_{\alpha'} ^{-1}) , (\varphi_{\beta} \circ \varphi_{\beta'} ^{-1} )$ are smooth (since, by assumption, the atlases of $M_{1}$ and $M_{2}$ consist of [[transition map|smoothly compatible]] charts). Hence $\mathscr{A}$ is smoothly compatible. > > Next we show that the $\pi_{i}$ are [[smooth maps between manifolds|smooth]] for $i=1,2$. WLOG $i=1$. Let $p=(p_{1},p_{2}) \in M_{1} \times M_{2}$ and take a [[coordinate chart]] $(U_{1} \times U_{2}, \varphi_{1} \times \varphi_{2})$ about $p$, where $(U_{1}, \varphi_{1})$ is a corresponding [[coordinate chart]] about $p_{1}=\pi_{1}(p) \in M_{1}$. We have $\begin{align} > \varphi_{1} \circ \pi_{1} \circ (\varphi_{1} \times \varphi_{2})^{-1} &=\varphi_{1} \circ \pi_{1} \circ (\varphi_{1}^{-1} \times \varphi_{2}^{-1}) \\ > &= \varphi_{1} \circ \varphi_{1} ^{-1} \\ > &=\id_{\mathbb{R}^{m_{1}}}, > \end{align}$ > which is [[continuously differentiable|smooth]]. > > Finally, we check the characteristic property. > > $\to$. If $f:N \to M_{1} \times M_{2}$ is [[smooth maps between manifolds|smooth]], then so is $\pi_{i} \circ f$ as a composition of smooth maps. > > $\leftarrow.$ Suppose $\pi_{i} \circ f$ is smooth for $i=1,2$. $f$ must factor as $f_{1} \times f_{2}$ for $f_{1}:N \to M_{1}$ and $f_{2}:N \to M_{2}$; in fact, we must have $f_{1}=\pi_{1} \circ f$ and $f_{2}=\pi_{2} \circ f$. Thus the components of $f$ are each smooth and so $f$ is smooth as well. > [!basicnonexample] Warning. > Let $M$ be a [[smooth manifold]], $U \subset M$ a nonempty open subset and $f:U \to \mathbb{R}$ a smooth function. Then $f$ need not extend smoothly to all of $M$. As a counterexample, consider the chart $\sigma_{N}$ [[stereographic projection|stereographically projecting]] $\mathbb{S}^{1}-\{ N \}$, $N$ the north pole, onto $\mathbb{R}$. An smooth extension $f:\mathbb{S}^{1} \to \mathbb{R}$ of this map to all of $\mathbb{S}^{1}$ would in particular need to be [[continuous]], and hence commute with any [[function limit|function limit]] in $\mathbb{S}^{1}$ converging to $p$; however, the corresponding limit in $\mathbb{R}$ does not exist. ^nonexample ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```