In the final differential geometry basics post, we will define differentiable structure and prove some important results.
DEFINITION: Let be a topological manifold. If is a collection of charts such that the s cover , then we call an atlas for . If two charts have the property that the transition maps and are diffeomorphisms whenever , then we say that are compatible, or smoothly compatible. If all the charts in an atlas are smoothly compatible with one another, then we say is a smooth atlas.
In the above definition we can easily generalise to compatibility for any , 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 be a topological manifold, and let be a collection of charts. Then we say the pair is a smooth manifold if is a maximal smooth atlas for : 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 a structure on , or a smooth structure on .
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.
Lemma: Let be a topological manifold, and let be a smooth atlas for . Then is contained in a unique smooth structure .
Proof*: Define to be the collection of charts for which are smoothly compatible with every chart in . Obviously , because every chart in is trivially compatible with every chart in since it is a smooth atlas. Also, the domains of the charts in cover , since at least the charts in do. Hence is a smooth atlas. Getting there!
Now take , where . If we show these arbitrary charts are smoothly compatible, then we will have shown that is a smooth atlas. What we will show is that for every point , there exists an open neighbourhood of contained in upon which is smooth (i.e. the map is smooth at ). This will tell us in fact that is smooth on . So let’s get to work.
Let . Then, since is an atlas for , there exists some chart such that . By definition of , the charts and are smoothly compatible with . This means that the transition maps and are smooth. Since is open in , we have that is open, and hence open in Euclidean space. Similarly, and are open. So restricting the domains, we see that the maps and are smooth. Since the composition of smooth maps is smooth, we have that is smooth on . Hence is smooth at . Since was arbitrary, we have that is smooth on . Since we chose arbitrary charts, it follows that this map is a diffeomorphism.
It remains to show that is maximal, and unique. If we take a chart smoothly compatible with every chart in , then it must be smoothly compatible with each chart in . Hence by definition, . This proves maximality. For uniqueness suppose that is a smooth structure such that . Then it follows that every chart in is compatible with each chart in , and hence . But we assumed was maximal, so . Hence .
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 be a topological manifold, and let be two smooth atlases for . Then
is a smooth atlas.
PROOF: Suppose . Choose a chart . Then either, or . Without loss of generality, suppose that . Then it’s enough to show that is compatible with every chart in . So . But , so is compatible with every chart in . Hence is a smooth atlas.
Conversely, suppose that is a smooth atlas. This part of the argument is quite similar to the proof of the previous lemma. Let . Then is smoothly compatible with every chart in . We now want to show that is smoothly compatible with every chart in . Let be a chart in such that . Let . Then since is a smooth atlas, there exists such that (since the domains of the charts cover ). Since is smoothly compatible with , we have that is smooth at . By hypothesis, and are smoothly compatible, so is smooth at . The composition is therefore smooth at . In a similar way, is smooth at . Since every element in is of the form , and every element of is of the form , it follows that the transition maps and are smooth on and respectively. Hence is a diffeomorphism. Therefore the charts and are smoothly compatible. Since was arbitrary in , we have shown that . Hence . By a symmetric argument .
Anyway, that’s enough for today. In the next Differential Geometry post, we’ll discuss graphs and spheres.