If I’m going to take this renewed interest in trying to understand differential geometry more seriously, I might as well read more than one source on it. So, I’ll start a sequence of posts on this book as well.
Just as Merkulov, Lee starts with a short definition of what a topological manifold is, including definitions for the terms needed for the treatment.
An n-dimensional topological manifold is a second countable Haussdorff space of local Euclidean dimension n.
Next, Lee goes on to define coordinate charts. I won’t repeat the treatment, since he doesn’t really bring anything Merkulov hasn’t talked about in some manner or other. Atlases, smooth structures and equivalence of atlases also merits some treatment.
The first really new thing I find is the Lemma 1.4. Lee points out that for a topological manifold M, every smooth atlas is contained in a unique maximal smooth atlas – this is probably a rather straightforward application of Zorn’s lemma, but Lee gives an explicit construction of the unique maximal smooth atlas as theatlas of all charts that are smoothly compatible with every chart in the original atlas. There are some technicalities to be checked to verify that this is an atlas and that it is maximal; but it all boils down to “because it’s compatible anyway”.
Furthermore, for a second part he affirms that two smooth atlases determine the same maximal smooth atlas iff their union is a smooth atlas. The proof of this is left to the reader; but what would my blogposts be if not the reader doing the things he’s supposed to do? So here goes.
Suppose two smooth atlases determine the same maximal smooth atlas. Say and . We want to show that then their union is a smooth atlas. Obviously their union is an atlas, so we need only verify smoothness. So take some pair of charts and . We need to show that on the respective images of the functions and are smooth. So we pick some arbritrary point and want to show smoothness at this point.
But we can use the maximal smooth atlas now. We pick a chart such that . Since this chart is from the maximal smooth atlas, it is compatible with both . More precisely, this means that is smooth. So is also . So if we compose these two functions, we get a new function. That takes points in to points in . And this function is, as it is the composition of two smooth functions, also smooth.
For a different point we may have to pick a different chart from , but since all charts in are smoothly compatible with all charts in both and , this won’t really change much of the argument. And thus we know that if both determine the same maximal atlas then their union is a smooth atlas.
Suppose now that the two atlases and have a smooth atlas as their union. That means more specifically that all the charts in are smoothly compatible with all charts in , and thus that is contained in the maximal smooth atlas containing and vice versa. But since the maximal smooth atlas containing a specific atlas was unique, they both have the same maximal smooth atlas.
And this illuminates why the interesting condition Merkulov gave was precisely that they have a smooth union.
Note that any manifold that can be covered by a single chart thus has a smooth structure determined in its entirety by that chart.
And at this point – page 11 – Lee does something that I probably will end up hating him for. He introduces the Einstein Summation Convention. It’s bad. It’s ugly. And it stands for most of the things that led to my flunking differential geometry in the first place. Boooo!
Another few pages yields an idiotic definition and the introduction of diffeomorphisms without telling the reader what they are. I’ll stop reading this thing now.