If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
Contents
Statement
Let [ilmath]\mathcal{A}:\eq\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in I} [/ilmath] be smooth atlas on some locally euclidean topological space, [ilmath](X,\mathcal{J})[/ilmath]^{[Note 1]} and suppose [ilmath](V,\psi)[/ilmath] and [ilmath](W,\sigma)[/ilmath] are charts, then^{[1]}:
- If we have both:
- [ilmath](V,\psi)[/ilmath] is smoothly compatible with the atlas [ilmath]\mathcal{A} [/ilmath] and
- [ilmath](W,\sigma)[/ilmath] is also smoothly compatible with the atlas [ilmath]\mathcal{A} [/ilmath]
- then these two charts are themselves smoothly compatible with each other, i.e.:
- [ilmath](V,\psi)[/ilmath] and [ilmath](W,\sigma)[/ilmath] are smoothly compatible charts
Proof
We wish to show that [ilmath](V,\psi)[/ilmath] and [ilmath](W,\sigma)[/ilmath] are smoothly compatible charts, i.e. that the transition maps are smooth (in the usual [ilmath]\mathbb{R}^n[/ilmath] sense), that is to say that:
- [ilmath](\sigma\circ\psi^{-1}):\underbrace{\psi(V\cap W)}_{\subseteq\mathbb{R}^m}\rightarrow\underbrace{\sigma(V\cap W)}_{\subseteq\mathbb{R}^n} [/ilmath]^{[Note 2]} is smooth in the real analysis sense
- [ilmath](\psi\circ\sigma^{-1}):\sigma(V\cap W)\rightarrow\psi(V\cap W)[/ilmath] must also be smooth in the same sense (so we have a diffeomorphism)
Recall that to be considered smooth or [ilmath]C^\infty[/ilmath] they must be smooth at each point in the domain, we will show [ilmath]\forall p\in V\cap W[\sigma\circ\psi^{-1}\text{ is } [/ilmath][ilmath]\text{smooth} [/ilmath][ilmath]\text{ at }p][/ilmath], with this said we may now begin the proof:
Proof:
- If [ilmath]V\cap W[/ilmath] is empty then we're done, the statement is vacuously true
- Otherwise:
- Let [ilmath]p\in V\cap W[/ilmath] be given (as in this case there is a [ilmath]p[/ilmath] to speak of)
- As [ilmath]\mathcal{A} [/ilmath] is a smooth atlas we see that [ilmath]X\subseteq\bigcup_{\alpha\in I}U_\alpha[/ilmath]^{[Note 3]}
- By the implies-subset relation this is the same as:
- [ilmath]\forall x\in X[x\in \bigcup_{\alpha\in I}U_\alpha][/ilmath], then
- by the definition of union this is the same as (as in if and only if or [ilmath]\iff[/ilmath])
- [ilmath]\forall x\in X[\exists \beta\in I[x\in U_\beta]][/ilmath] which we can re-write more neatly as:
- [ilmath]\forall x\in X\exists \beta\in I[x\in U_\beta][/ilmath]
- by the definition of union this is the same as (as in if and only if or [ilmath]\iff[/ilmath])
- [ilmath]\forall x\in X[x\in \bigcup_{\alpha\in I}U_\alpha][/ilmath], then
- By the implies-subset relation this is the same as:
- Both [ilmath]V\subseteq X[/ilmath] and [ilmath]W\subseteq X[/ilmath] of course, so [ilmath]V\cap W\subseteq X[/ilmath]^{[Note 4]}
- We see that [ilmath]p\in X[/ilmath] as a result (we already knew this but just to as formal as usual)
- Thus: [ilmath]\exists \beta\in I[p\in U_\beta][/ilmath]
- Define [ilmath]\beta\in I[/ilmath] to be such that [ilmath]p\in U_\beta[/ilmath] and [ilmath]\varphi_\beta[/ilmath] is a chart containing [ilmath]p[/ilmath]
- Now [ilmath]p\in U_\beta\cap V\cap W[/ilmath]
- By hypothesis the following four maps are smooth on their domains (by compatibility with the atlas):
- [ilmath](\psi\circ\varphi_\beta^{-1}):\varphi_\beta(V\cap U_\beta)\rightarrow\psi(V\cap U_\beta)[/ilmath] - which is one direction of the transition maps between [ilmath](V,\psi)[/ilmath] and [ilmath](U_\beta,\varphi_\beta)[/ilmath]
- [ilmath](\varphi_\beta\circ\psi^{-1}):\psi(V\cap U_\beta)\rightarrow\varphi_\beta(V\cap U_\beta)[/ilmath] - which is the other direction of the transition maps between [ilmath](V,\psi)[/ilmath] and [ilmath](U_\beta,\varphi_\beta)[/ilmath]
- [ilmath](\sigma\circ\varphi_\beta^{-1}):\varphi_\beta(W\cap U_\beta)\rightarrow\sigma(W\cap U_\beta)[/ilmath] - which is one direction of the transition maps between [ilmath](W,\sigma)[/ilmath] and [ilmath](U_\beta,\varphi_\beta)[/ilmath]
- [ilmath](\varphi_\beta\circ\sigma^{-1}):\sigma(W\cap U_\beta)\rightarrow\varphi_\beta(W\cap U_\beta)[/ilmath] - which is the other direction of the transition maps between [ilmath](W,\sigma)[/ilmath] and [ilmath](U_\beta,\varphi_\beta)[/ilmath]
- As [ilmath]\mathcal{A} [/ilmath] is a smooth atlas we see that [ilmath]X\subseteq\bigcup_{\alpha\in I}U_\alpha[/ilmath]^{[Note 3]}
- Let [ilmath]p\in V\cap W[/ilmath] be given (as in this case there is a [ilmath]p[/ilmath] to speak of)
The message provided is:
- [ilmath](\psi\circ\varphi_\beta^{-1})\big\vert_{\varphi_\beta(V\cap W\cap U_\beta)}\circ(\varphi_\beta\circ\sigma^{-1})\big\vert_{\sigma(V\cap W\cap U_\beta)} [/ilmath] - notice the restrictions in play - as this composition is not defined, but can be defined on a subset of the domain of each part!
- This requires us to show that [ilmath]\left((\varphi_\beta\circ\sigma^{-1})\big\vert_{\sigma(V\cap W\cap U_\beta)}\right)(\sigma(V\cap W\cap U_\beta))\subseteq\varphi_\beta(V\cap W\cap U_\beta)[/ilmath] for the composition of the restrictions to be defined
- Then we say:
- [ilmath](\psi\circ\varphi_\beta^{-1})\big\vert_{\varphi_\beta(V\cap W\cap U_\beta)}\circ(\varphi_\beta\circ\sigma^{-1})\big\vert_{\sigma(V\cap W\cap U_\beta)} [/ilmath]
- [ilmath]\eq(\psi\circ\sigma^{-1})\big\vert_{\sigma(V\cap W\cap U_\beta)} [/ilmath]
- [ilmath](\psi\circ\varphi_\beta^{-1})\big\vert_{\varphi_\beta(V\cap W\cap U_\beta)}\circ(\varphi_\beta\circ\sigma^{-1})\big\vert_{\sigma(V\cap W\cap U_\beta)} [/ilmath]
- Then as [ilmath](\varphi_\beta\circ\sigma^{-1})[/ilmath] and [ilmath](\psi\circ\varphi_\beta^{-1})[/ilmath] are smooth (by hypothesis)
- so are their restrictions TODO: Are they though?
- Thus so is [ilmath](\psi\circ\sigma^{-1})\big\vert_{\sigma(V\cap W\cap U_\beta)} [/ilmath] - specifically it is smooth at [ilmath]\sigma(p)[/ilmath]
- If a map is smooth at a point in its domain then any extension of that map is smooth at that point TODO: Prove this too!
- Thus we see [ilmath](\psi\circ\sigma^{-1})[/ilmath] is smooth at [ilmath]p[/ilmath]
- If a map is smooth at a point in its domain then any extension of that map is smooth at that point
- so are their restrictions
- Since [ilmath]p[/ilmath] was arbitrary we see the transition maps are smooth everywhere.
The other transition map is just the composition of the remaining 2 smooth transition maps we get by hypothesis and is basically the same work.
References
Notes
- ↑ This page's statement is used to build a maximal smooth atlas and thus a smooth manifold from a topological manifold. We may speak of a (smooth) atlas on a space that is simply locally euclidean, it need not be a full on topological manifold so we relax that constraint
- ↑ Notice that we speak of [ilmath]\mathbb{R}^m[/ilmath] and [ilmath]\mathbb{R}^n[/ilmath], this is because the dimension of connected components may vary from component to component, in this case though [ilmath]V\cap W[/ilmath] would be empty of course.
- ↑ We actually have [ilmath]X\eq\bigcup_{\alpha\in I}U_\alpha[/ilmath] as the [ilmath]U_\alpha[/ilmath] are subsets of [ilmath]X[/ilmath] (in fact open sets so [ilmath]\in\mathcal{J} [/ilmath]) so we have [ilmath]\bigucp_{\alpha\in I}U_\alpha\subseteq X[/ilmath] automatically, we use [ilmath]X\subseteq\bigcup_{\alpha\in I}U_\alpha[/ilmath] for the atlas property to use the implies-subset relation to emphasise what we're doing. As we have just shown this is logically the same as equality between [ilmath]X[/ilmath] and the union
- ↑ TODO: What claim have I used here? The intersection of sets is a subset of each set might be related (or indeed good enough as then [ilmath]V\cap W\subseteq V\subseteq X[/ilmath]!)
- XXX Todo
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