---- > [!definition] Definition. ([[diffeomorphism]]) > If $M$ and $N$ are [[smooth manifold|smooth manifolds]] (with or without boundary), a **diffeomorphism from $M$ to $N$** is a [[smooth maps between manifolds|smooth]] [[bijection|bijective]] map $F:M \to N$ that has a [[smooth maps between manifolds|smooth]] inverse. We call $M,N$ **diffeomorphic**. > \ > > [!note] Remark. > Two manifolds with inequivalent [[smooth structure|smooth structures]] may nevertheless be diffeomorphic. > > For example, cover $\mathbb{R}$ as a set first with the (single) map $\varphi_{1}(x)=x$ and with the (single) map $\varphi_{2}(x)=x^{3}$ ($x \in \mathbb{R}$). Since the Euclidean function $\varphi_{1} \circ \varphi_{2}^{-1}(x)=x^{1/3}$ is not [[continuously differentiable|smooth]] at the origin, $\varphi_{1}$ and $\varphi_{2}$ must not belong to the same $C^{\infty}$ structure on $\mathbb{R}$. > > However, the [[smooth manifold|smooth manifolds]] $R_{1},R_{2}$ with topology induced by $\varphi_{1}, \varphi_{2}$ respectively, are diffeomorphic. The diffeomorphism is just $\begin{align} > h : R_{1} & \to R_{2} \\ > x & \mapsto x^{3} > \end{align}$ > with inverse $\begin{align} > g: R_{1} &\to R_{2} \\ > y &\mapsto y^{1/3} > \end{align}$ > since we have $\varphi_{1} \circ h \circ \varphi_{2}^{-1}(x)=\varphi \circ h(x^{1/3})=\varphi_{1}(x)=x$ and $\varphi_{2} \circ g \circ \varphi_{1} ^{-1}(y)=\varphi_{2} \circ g(y)= \varphi_{2}(y^{1/3})=y,$ meaning that $\varphi_{1} \circ h \circ \varphi_{2} ^{-1}$ and $\varphi_{2} \circ g \circ \varphi_{1}$ are just the [[identity map]] on $\mathbb{R}$ and hence smooth. > [!justification] Motivation. > Diffeomorphisms are [[homeomorphism|homeomorphisms]] that also preserve the differential structures of manifolds, in the sense that a function/map $G$ defined on $N$ is smooth if and only if the function/map $G \circ F$ is too. > [!basicexample] > The map from the open upper [[cone of a topological space]] $S=\{ \}$ [[TODO]] > The [[ellipsoid]] $\mathbb{E}^{2} := \left\{ (x,y,z) \in \mathbb{R}^{3} : \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1 \right\},$ and [[sphere]] $\mathbb{S}^{2} := \{ (x,y,z) \in \mathbb{R}^{3}: x^{2}+y^{2}+z^{2} = 1 \}$in $\mathbb{R}^{3}$ are [[diffeomorphism|diffeomorphic]] as [[differentiable Euclidean submanifold|regular surfaces]]. We claim the map $\begin{align} F:\mathbb{S}^{2} & \to \mathbb{E}^{2} \\ (x,y,z) & \mapsto (ax, by, cz) \end{align}$ is a [[diffeomorphism]] witnessing this. First we check that $F$ is actually a map from $\mathbb{S}^{2}$ to $\mathbb{E}^{2}$: if $(x,y,z) \in \mathbb{S}^{2}$ then $\begin{align} \frac{F_{1}^{2}(x,y,z)^{}}{a^{2}}+ \frac{F_{2}^{2}(x,y,z)}{b^{2}}+\frac{F_{3}^{2}(x,y,z)}{c^{2}}= & \frac{(ax)^{2}}{a^{2}} + \frac{(by)^{2}}{b^{2}} + \frac{(cz)^{2}}{c^{2}} \\ = & x^{2} + y ^{2} + z^{2} \\ = & 1, \end{align}$ and so $F(x,y,z) \in \mathbb{E}^{2}$. So indeed $F(\mathbb{S}^{2}) \subset \mathbb{E}^{2}$. > Now define $\begin{align} G: \mathbb{E}^{2} & \to \mathbb{S}^{2} \\ (x,y,z) & \mapsto \left( \frac{x}{a}, \frac{y}{b}, \frac{z}{c} \right); \end{align}$ note that $G$ is indeed a map from $\mathbb{E}^{2}$ to $\mathbb{S}^{2}$ because $\begin{align} G_{1}^{2}(x,y,z) + G_{2}^{2}(x,y,z) + G_{3}^{2}(x,y,z)= & \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}+ \frac{z^{2}}{c^{2}} \\ = & 1. \end{align}$ Moreover, $G=F^{-1}$ because $\begin{align} (G \circ F)(x,y,z)= & G( (ax,by, cz) ) \\ = & \left( \frac{ax}{a} , \frac{bx}{b}, \frac{cx}{c} \right)\\ = & (x,y,z) \\ = & \left( a\left( \frac{x}{a} \right), b\left( \frac{y}{b} \right), c \left( \frac{z}{c} \right) \right) \\ = & F \left( \frac{x}{a}, \frac{y}{b}, \frac{z}{c} \right) \\ = & (F \circ G)(x,y,z). \end{align}$ $F$ and $F^{-1}$ are each [[smooth maps between manifolds|smooth]] because they are [[restrictions of smooth maps to regular surfaces are smooth|restrictions]] of the [[smooth]] (linear) dilation maps $(x,y,z) \mapsto (ax,by,cz)$ and $(x,y,z)\mapsto (\frac{x}{a}, \frac{y}{b}, \frac{z}{c})$ respectively. ^57b3b4 ---- #### ---- #### 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 > ```