Timeline for How much of differential geometry can be developed entirely without atlases?
Current License: CC BY-SA 2.5
10 events
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| Jun 17, 2010 at 14:55 | comment | added | Sean Tilson | @orby: this is a question i might well have asked. I understand that hard analysis is important, but i am not a hard analyst and i would still like to have some understanding. For myself at least, things like vector bundles are easier for me to understand than some relation between christoffel symbols, its just the way i think. I am particularly interested in things that deane seems to be alluding to above, global geometric type results from PDE! | |
| Feb 11, 2010 at 2:42 | comment | added | Deane Yang | I don't really follow the debate going on here. So what if a few key tools such as the IFT need to be proved using local co-ordinates. Once you have it, it can be stated and used in a co-ordinate-free environment. Virtually all of PDE theory is proved using local or global co-ordinates, but once you have developed the necessary tools and restated the theorems in a co-ordinate-free form, you can often successfully avoid co-ordinates when applying the PDE theorems to differential geometry. | |
| Feb 10, 2010 at 11:41 | comment | added | Orbicular | Andrew, I have to agree with you. What's the point of doing differential geometry without a being able to do analysis on them? I enjoy synthetic reasoning, but when looking at how far they got, I'd rather rely on a theory with an inverse function theorem. (Anyone ever thought about sheaves with values in Hilbert/Banach manifolds?) | |
| Feb 10, 2010 at 11:04 | comment | added | Andrew Stacey | This reminded me of the message that I try to get across in my "comparative smootheology" talks (that's talks, not article): that the point (ha ha) of smooth manifolds is to define what "smooth map" means and for that we only need to agree on what smooth curves are - the whole "atlas" and "chart" stuff is just to ensure that we agree on this. Interestingly, if you read Kriegl and Michor's book then you get the impression that they regard IFT as being what separates "hard" and "soft" geometry (my words). | |
| Feb 10, 2010 at 10:15 | comment | added | Harry Gindi | Yeah, I'm sure he knows what he's doing too. This is not some attempt to "prove him wrong", but rather to find a reference that discusses all of this in a more palatable way for me, at least. | |
| Feb 10, 2010 at 10:09 | comment | added | Pete L. Clark | Hmm. I think I would rather say that the proof would be word for word identical in the language of LRS [except for removing all instances of "atlas" from the proof; I would be mildly surprised to find any]. But I'm not teaching the class and I don't want to contradict the professor, who I'm sure knows what he is doing. | |
| Feb 10, 2010 at 9:59 | comment | added | Harry Gindi | Well, a few days ago in my differential geometry class, a proof was given of some fact (I think it was called the "local form of an immersion" in book I of spivak). I asked the professor if there was an analogue of the proof stated in terms of locally ringed spaces, and he said that the proof relied on the IFT, so there was no natural way to phrase it in terms of LRSpaces. This question was kinda of a "half-question" and "half-reference-request" | |
| Feb 10, 2010 at 9:52 | comment | added | Pete L. Clark | I actually don't see how atlases come up in the statement of the inverse function theorem [or even what this result has to do with manifolds at all, really; you can apply it to maps between manifolds, but by nature it's a local result]. Do you have a specific formulation or reference in mind? | |
| Feb 10, 2010 at 9:45 | comment | added | Harry Gindi | That's what I'd hoped to hear. The question is: things like the inverse function theorem seem like they're more easily stated using atlases (at least I don't know of a good statement using locally ringed spaces). I'd like to see all of the common theorems stated in terms of LRS's, whence came the book request. | |
| Feb 10, 2010 at 9:39 | history | answered | Pete L. Clark | CC BY-SA 2.5 |