Timeline for How much of differential geometry can be developed entirely without atlases?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Feb 10, 2010 at 11:36 | comment | added | Orbicular | Well, you don't really remove the maximal atlas, you just hide it in another box. It's fairly impossible to check whether an atlas is maximal, that's obviously true. But on the other hand - do you know all open sets of R^n, together with all smooth functions on them? In practice you are dealing with an open cover only (Cech-style). The same is true for atlases! | |
| Feb 10, 2010 at 10:58 | comment | added | Harry Gindi | Rather, the analytic structure associated with the variety. | |
| Feb 10, 2010 at 10:34 | comment | added | Harry Gindi | And of course I've "tried it!" it was on homework 2 or 3. | |
| Feb 10, 2010 at 10:33 | comment | added | Harry Gindi | I've put an explanation in the OP. Proving equivalence is pretty straightforward. The problem is, sometimes proving something about a mathematical object is easier in one formalism than another one. For example, the functor of points approach to algebraic geometry allows us to get lots of proofs for free, but some proofs are still easier using locally ringed spaces (consider the algebraic and analytic structures of a complex variety). | |
| Feb 10, 2010 at 10:20 | comment | added | Pete L. Clark | @Kevin: Yes, that's exactly what I was getting at in my answer. +1 for your exhortation "Try it!" which would be my advice as well. | |
| Feb 10, 2010 at 10:15 | history | undeleted | Kevin H. Lin | ||
| Feb 10, 2010 at 10:10 | history | deleted | Kevin H. Lin | ||
| Feb 10, 2010 at 10:10 | history | answered | Kevin H. Lin | CC BY-SA 2.5 |