Timeline for How much of differential geometry can be developed entirely without atlases? [closed]
Current License: CC BY-SA 2.5
32 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 18, 2016 at 13:19 | comment | added | Stefan Kohl♦ | I see 3 reopen votes, but no related (or any other recent) comment. | |
| Apr 18, 2016 at 11:16 | review | Reopen votes | |||
| Apr 18, 2016 at 13:17 | |||||
| Aug 4, 2015 at 19:24 | comment | added | user21349 | related: math.stackexchange.com/questions/53021/… | |
| Aug 14, 2011 at 8:44 | comment | added | Marc Palm | Alain Connes has shown, that you can encode the structure of a Riemannian manifold in a spectral triple. You should think about it as an analogue of Gelfand-Naimark for topological spaces. | |
| Jan 15, 2011 at 14:27 | history | closed |
Harry Gindi Akhil Mathew user9198 Andrey Rekalo Qiaochu Yuan |
no longer relevant | |
| Jan 15, 2011 at 8:04 | comment | added | Tilman | This may be nitpicking, but OP's definition of a smooth manifold is actually inaccurate. The local homeomorphisms with $\mathbb{R}^n$ are part of the structure of a manifold, it's not enough just to require their existence. Otherwise you wouldn't be able to know how to pull the differentiable structure on $\mathbb{R}^n$ back to the manifold. (Remember there's more than one differentiable structure on $\mathbb{R}^n$.) So the atlas is already non-avoidable in the definition! | |
| Jan 4, 2011 at 18:44 | comment | added | Dick Palais | @Deane, @Igor: There is another approach that many geometers feel is as intrinsic as the "coordinate-free" approach and often makes calculations even more revealing and easier than "choosing well-adapted coordinates". I mean the "moving frame" approach, going back to Elie Cartan, and the favored method of Chern and his students. The point is that there is very frequently an absolutely natural choice of moving frame in a given situation (for example, for a manifold embedded in Euclidean space there is the principal curvature framing) and working in this makes many computations much clearer. | |
| Jan 4, 2011 at 10:32 | answer | added | Patrick I-Z | timeline score: 10 | |
| Oct 15, 2010 at 6:43 | answer | added | user9280 | timeline score: 1 | |
| Sep 11, 2010 at 12:43 | answer | added | Martin Gisser | timeline score: -1 | |
| Mar 4, 2010 at 0:56 | answer | added | BMann | timeline score: 1 | |
| Feb 10, 2010 at 19:22 | comment | added | Igor Belegradek | @Deane, when I was talking of coordinates I meant the frame in which the computation is done, e.g. for the complex projective space the way the frame is chosen should reflect the fact how the space came about, say as the base of a Riemannian submersion from a sphere. But you are right, there are other ways of doing this; I just do not find them easier. | |
| Feb 10, 2010 at 19:11 | comment | added | Igor Belegradek | @Harry: there are many cases when working in coordinates makes things much easier; in other situations coordinate-free arguments work better. I think sticking to either approach will inevitably "obscure important points". | |
| Feb 10, 2010 at 18:55 | answer | added | Emerton | timeline score: 27 | |
| Feb 10, 2010 at 18:41 | comment | added | Pete L. Clark | It seems now that the question was supposed to be "How much of differential geometry can be done in a global, coordinate-free way?" To me, this is significantly different from what was actually asked: atlases and LRSs use local coordinates in almost exactly the same way, I think. Maybe a revised title/revised question/further question would be helpful. | |
| Feb 10, 2010 at 18:37 | comment | added | Deane Yang | @Igor: I would agree that trying to study a Riemannian manifold via its sheaf of smooth functions has limited potential, but on the other hand, I do think a lot of Riemannian geometry is best done without co-ordinates. In co-ordinates, you have a lot of extra baggage that is a pain in the neck, including Christoffel symbols. As for your example of complex projective space, there are definitely clean and easy ways to compute its curvature without using co-ordinates. | |
| Feb 10, 2010 at 18:29 | comment | added | Ilya Grigoriev | The classical point of view used neither atlases nor locally ringed spaces. Instead, it was all about submanifolds of $R^n$, defined by functions. (Admittedly, one does need to prove a condition on derivatives of the functions that's equivalent to the submanifold being locally diffeomorphim to $R^m$). Riemann and Gauss got quite far with it. | |
| Feb 10, 2010 at 18:12 | answer | added | user717 | timeline score: 2 | |
| Feb 10, 2010 at 17:25 | comment | added | Harry Gindi | It's like in linear algebra, where choosing coordinates can actually obscure the important points. | |
| Feb 10, 2010 at 17:23 | comment | added | Harry Gindi | @Igor: I prefer to leave computations as the "last step", so to speak. That is, proofs should be coordinate free when possible. | |
| Feb 10, 2010 at 16:47 | comment | added | Igor Belegradek | I am a Riemannian geometer, and I am curious what you are trying to accomplish. The basic reason why some differential geometry is done in coordinates is that using the correct coordinates adapted for the problem at hand makes computations much easier and gives invaluable insight like the Jordan normal form gives insight into matrices. Try computing curvature tensor of complex projective space without coordinates. I know only a few cases where sheaf theoretic language helps but the the most part I do not see why bother. It is more efficient to use other methods. | |
| Feb 10, 2010 at 11:49 | vote | accept | Harry Gindi | ||
| Feb 10, 2010 at 10:22 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 180 characters in body; added 9 characters in body; added 9 characters in body
|
| Feb 10, 2010 at 10:22 | answer | added | Dmitri Pavlov | timeline score: 15 | |
| Feb 10, 2010 at 10:10 | answer | added | Kevin H. Lin | timeline score: 10 | |
| Feb 10, 2010 at 10:03 | answer | added | Dmitri Pavlov | timeline score: 12 | |
| Feb 10, 2010 at 9:59 | answer | added | babubba | timeline score: 32 | |
| Feb 10, 2010 at 9:39 | answer | added | Pete L. Clark | timeline score: 15 | |
| Feb 10, 2010 at 9:02 | answer | added | Orbicular | timeline score: 14 | |
| Feb 10, 2010 at 8:52 | comment | added | Shizhuo Zhang | Are you looking for some formal differential geometry? A.Rosenberg ever wrote some papers on this topic(but never published). He used noncommutative AG machine reconstruct the differential geometry. | |
| Feb 10, 2010 at 8:51 | history | edited | Harry Gindi | CC BY-SA 2.5 |
added 71 characters in body
|
| Feb 10, 2010 at 8:41 | history | asked | Harry Gindi | CC BY-SA 2.5 |