Differential Geometry Basics (III)

In the final differential geometry basics post, we will define differentiable structure and prove some important results.

DEFINITION: Let M be a topological manifold. If \mathcal{A}=\{ (U_{\alpha},\varphi_{\alpha})\}_{\alpha\in A} is a collection of charts such that the U_{\alpha}'s cover M, then we call \mathcal{A} an atlas for M. If two charts (U, \varphi),(V,\vartheta) \in \mathcal{A} have the property that the transition maps \vartheta \circ \varphi^{-1} and \varphi \circ \vartheta^{-1} are diffeomorphisms whenever U\cap V \not= \emptyset, then we say that (U,\varphi),(V,\vartheta) are C^{\infty} compatible, or smoothly compatible. If all the charts in an atlas \mathcal{A} are smoothly compatible with one another, then we say \mathcal{A} is a smooth atlas.

In the above definition we can easily generalise to C^k compatibility for any k\in\mathbb{N}, but we will be interested in smooth compatibility here. Now here is the massively important definition that took 3 posts to arrive at:

DEFINITION: Let M be a topological manifold, and let \mathcal{A} be a collection of charts. Then we say the pair (M,\mathcal{A}) is a smooth manifold if \mathcal{A} is a maximal smooth atlas for M: that is to say, it is the largest possible collection of charts which are smoothly compatible with each other. We call the maximal smooth atlas \mathcal{A} a C^{\infty} structure on M, or a smooth structure on M.

Finally! The next few posts in this category will be concerned with the meticulous examination of several examples of smooth manifolds, but first we will look at some results regarding the uniqueness of smooth structures.

Wrong kind of atlas…

Lemma: Let M be a topological manifold, and let \mathcal{A} be a smooth atlas for M. Then \mathcal{A} is contained in a unique smooth structure \overline{\mathcal{A}}.

Proof*: Define \overline{\mathcal{A}} to be the collection of charts for M which are smoothly compatible with every chart in \mathcal{A}. Obviously \mathcal{A} \subseteq \overline{\mathcal{A}}, because every chart in \mathcal{A} is trivially compatible with every chart in \mathcal{A} since it is a smooth atlas. Also, the domains of the charts in \overline{\mathcal{A}} cover M, since at least the charts in \mathcal{A} do. Hence \overline{\mathcal{A}} is a smooth atlas. Getting there!

Now take (U, \varphi), (V,\vartheta) \in \overline{\mathcal{A}}, where U\cap V \not= \emptyset. If we show these arbitrary charts are smoothly compatible, then we will have shown that \overline{\mathcal{A}} is a smooth atlas. What we will show is that for every point p \in \varphi(U\cap V), there exists an open neighbourhood of p contained in \varphi(U \cap V) upon which \vartheta \circ \varphi^{-1} is smooth (i.e. the map is smooth at p). This will tell us in fact that \vartheta \circ \varphi^{-1} is smooth on \varphi(U\cap V). So let’s get to work.

Let p=\varphi(x) \in \varphi(U \cap V). Then, since \mathcal{A} is an atlas for M, there exists some chart (W, \psi)\in\mathcal{A} such that x \in W. By definition of \overline{\mathcal{A}}, the charts (U,\varphi) and (V, \vartheta) are smoothly compatible with (W, \psi). This means that the transition maps \psi \circ \varphi^{-1}: \varphi (U \cap W) \rightarrow \psi (U \cap W) and \vartheta \circ \psi^{-1} : \psi (V \cap W) \rightarrow \vartheta(V \cap W) are smooth. Since U\cap V\cap W is open in U, we have that \varphi(U \cap V \cap W) \subseteq \varphi(U) is open, and hence open in Euclidean space. Similarly, \vartheta(U \cap V\cap W) and \psi(U \cap V \cap W) are open. So restricting the domains, we see that the maps \psi \circ \varphi^{-1}: \varphi (U \cap V \cap W) \rightarrow \psi (U \cap V \cap W) and \vartheta \circ \psi^{-1} : \psi (U \cap V \cap W) \rightarrow \vartheta(U \cap V \cap W) are smooth. Since the composition of smooth maps is smooth, we have that \vartheta \circ \varphi^{-1} \circ \varphi \circ \varphi^{-1} = \vartheta \circ \phi^{-1} is smooth on \varphi(U \cap V \cap W). Hence \vartheta \circ \varphi^{-1} is smooth at p. Since p \in \varphi ( U \cap V) was arbitrary, we have that \vartheta \circ \varphi^{-1} is smooth on \varphi(U \cap V). Since we chose arbitrary charts, it follows that this map is a diffeomorphism.

It remains to show that \overline{\mathcal{A}} is maximal, and unique. If we take a chart (U, \varphi) smoothly compatible with every chart in \overline{\mathcal{A}}, then it must be smoothly compatible with each chart in \mathcal{A}. Hence by definition, (U,\varphi)\in\overline{\mathcal{A}}. This proves maximality. For uniqueness suppose that \mathcal{B} is a smooth structure such that \mathcal{A} \subseteq \mathcal{B}. Then it follows that every chart in \mathcal{B} is compatible with each chart in \mathcal{A}, and hence \mathcal{B} \subseteq \overline{\mathcal{A}}. But we assumed \mathcal{B} was maximal, so \overline{\mathcal{A}}\subseteq\mathcal{B}. Hence \mathcal{B} = \overline{\mathcal{A}}. \Box

This is a very powerful result. It tells us that we only have to define a smooth atlas on a topological manifold (which can be easy to do) and then we get a unique smooth structure. We won’t often know what all the elements are in the maximal atlas (because it can be absolutely massive), but it’s important to know such a thing exists. The following Theorem is particularly useful.

THEOREM: Let M be a topological manifold, and let \mathcal{A},\mathcal{B} be two smooth atlases for M. Then

\overline{\mathcal{A}} = \overline{\mathcal{B}} \iff \mathcal{A}\cup\mathcal{B} is a smooth atlas.

PROOF: Suppose \overline{\mathcal{A}} = \overline{\mathcal{B}}. Choose a chart (U,\varphi) \in \mathcal{A}\cup\mathcal{B}. Then either, (U,\varphi) in \mathcal{A} or (U,\varphi) \in \mathcal{B}. Without loss of generality, suppose that (U, \varphi) \in \mathcal{B}. Then it’s enough to show that (U, \varphi) is compatible with every chart in \mathcal{A}. So (U, \varphi) \in \mathcal{B} \implies (U, \varphi)\in \overline{\mathcal{B}}. But \overline{\mathcal{B}}=\overline{\mathcal{A}}, so (U,\varphi) is compatible with every chart in \mathcal{A}. Hence \mathcal{A}\cup\mathcal{B} is a smooth atlas.

Conversely, suppose that \mathcal{A}\cup\mathcal{B} is a smooth atlas. This part of the argument is quite similar to the proof of the previous lemma. Let (U,\varphi) \in \overline{\mathcal{A}}. Then (U,\varphi) is smoothly compatible with every chart in \mathcal{A}. We now want to show that (U,\varphi) is smoothly compatible with every chart in \mathcal{B}. Let (V,\vartheta) be a chart in \mathcal{B} such that U \cap V \not= \emptyset. Let x \in U \cap V. Then since \mathcal{A} is a smooth atlas, there exists (W,\psi) \in \mathcal{A} such that x \in W (since the domains of the charts cover M). Since (U,\varphi) is smoothly compatible with (W,\psi), we have that \psi\circ \varphi^{-1} is smooth at \varphi(x). By hypothesis, (V,\vartheta) and (W, \psi) are smoothly compatible, so \vartheta \circ \psi^{-1} is smooth at \psi(x). The composition \vartheta \circ \varphi^{-1} is therefore smooth at \varphi(x). In a similar way, \varphi\circ\vartheta^{-1} is smooth at \vartheta(x). Since every element in \varphi(U \cap V) is of the form \varphi(x), and every element of \vartheta(U \cap V) is of the form \vartheta(x), it follows that the transition maps \vartheta\circ\varphi^{-1} and \varphi\circ\vartheta^{-1} are smooth on \varphi(U \cap V) and \vartheta (U \cap V) respectively. Hence \vartheta\circ\varphi^{-1} is a diffeomorphism. Therefore the charts (U, \varphi) and (V, \vartheta) are smoothly compatible. Since (V, \vartheta) was arbitrary in \mathcal{B}, we have shown that (U, \varphi) \in \overline{\mathcal{B}}. Hence \overline{\mathcal{A}}\subseteq \overline{\mathcal{B}}. By a symmetric argument \overline{\mathcal{B}}\subseteq \overline{\mathcal{A}}. \Box

Wrong kind of smooth…

Anyway, that’s enough for today. In the next Differential Geometry post, we’ll discuss graphs and spheres.

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About Paul

I am primarily a mathematician, secondarily a musician and fundamentally a connoisseur of delicious foodstuffs. I have a BSc in Mathematics from the University of Glasgow and MASt in Mathematics from the University of Cambridge. I am currently a PhD student in categorical algebra at Glasgow.
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