Timeline for The purpose of connections in differential geometry [closed]
Current License: CC BY-SA 4.0
37 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2021 at 10:29 | history | closed |
Alex M. Alexey Ustinov Stefan Waldmann Chris Gerig Johannes Hahn |
Not suitable for this site | |
| Sep 14, 2021 at 7:56 | review | Close votes | |||
| Sep 16, 2021 at 10:29 | |||||
| Sep 10, 2021 at 11:50 | comment | added | alvarezpaiva | Think of "classical geometry" as the study of congruences of figures in a homogeneous space F. Now assume you have not just one, but many copies of F indexed by a parameter x lying on a manifold B. In general, you can't compare things in Fx with things in Fy if x is different from y. You need extra-structure. That structure is a connection, which allows you to compare what is going in both spaces provided you have a path in B from x to y. | |
| Sep 5, 2021 at 16:26 | answer | added | Ben McKay | timeline score: 0 | |
| Sep 5, 2021 at 15:56 | history | reopened |
user44143 Ryan Budney Will Sawin Greg Friedman Turbo |
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| S Sep 3, 2021 at 14:45 | vote | accept | Canrif | ||
| Sep 3, 2021 at 13:31 | comment | added | Michael Engelhardt | @AlexandreEremenko - ah, now I understand what you mean. I just had trouble finding the right "meta"-level of discussion. I agree, yes. | |
| S Sep 3, 2021 at 13:28 | vote | accept | Canrif | ||
| S Sep 3, 2021 at 14:45 | |||||
| Sep 3, 2021 at 13:26 | comment | added | Canrif | @AlexandreEremenko to be clear, I'm not asking what "the purpose of differential geometry" is I just want an example of it being used so I can understand the kinds of problems it helps with. | |
| Sep 3, 2021 at 12:12 | comment | added | Alexandre Eremenko | @Michael Engelgardt: No I do not doubt that DG is necessary for GR:-) I just wanted to say that the question about "purpose of DG" is meaningless. | |
| S Sep 3, 2021 at 10:23 | vote | accept | Canrif | ||
| S Sep 3, 2021 at 13:28 | |||||
| S Sep 3, 2021 at 10:23 | vote | accept | Canrif | ||
| S Sep 3, 2021 at 10:23 | |||||
| S Sep 3, 2021 at 10:23 | vote | accept | Canrif | ||
| S Sep 3, 2021 at 10:23 | |||||
| Sep 3, 2021 at 10:22 | vote | accept | Canrif | ||
| S Sep 3, 2021 at 10:23 | |||||
| S Sep 3, 2021 at 7:15 | review | Reopen votes | |||
| Sep 5, 2021 at 15:58 | |||||
| S Sep 3, 2021 at 7:15 | history | edited | user44143 | CC BY-SA 4.0 |
focused the question on connections, since the upvoted answers already focused on that topic
Added to review
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| Sep 3, 2021 at 6:17 | history | closed |
abx Ben McKay LSpice Alexandre Eremenko alvarezpaiva |
Needs more focus | |
| Sep 3, 2021 at 5:09 | answer | added | bubba | timeline score: 3 | |
| Sep 3, 2021 at 2:25 | comment | added | Michael Engelhardt | @AlexandreEremenko - I'm not sure I'm able to parse the syntax of your remark. Black holes here stand pars pro toto for General Relativity. Are you doubting that we need Differential Geometry to understand General Relativity? Or are you doubting that General Relativity is interesting? That GPS is useful? | |
| Sep 3, 2021 at 1:23 | comment | added | Alexandre Eremenko | @Michael Engelgardt: This does not justify DG: why do we want to understand black holes? | |
| Sep 2, 2021 at 22:36 | history | became hot network question | |||
| Sep 2, 2021 at 20:50 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Sep 2, 2021 at 18:55 | answer | added | Gabe K | timeline score: 4 | |
| Sep 2, 2021 at 18:25 | answer | added | Ryan Budney | timeline score: 7 | |
| Sep 2, 2021 at 18:08 | answer | added | Liviu Nicolaescu | timeline score: 7 | |
| Sep 2, 2021 at 17:35 | comment | added | Willie Wong | Of the three you listed: Connexion is the easiest to explain. We need a connection as soon as we want to compare things that are situated at different points on a manifold. Imagine two kids on the playground each pointing a finger. How would you decide if they are pointing in the same direction? (The unfortunately thing is that it turns out how you compare things depends on the path you take to get from point A to point B.) I have some notes about this which may be too technical. But maybe it helps a little. | |
| Sep 2, 2021 at 15:57 | comment | added | Daniel Waters | @user347489 another one would be computer graphics, as discrete dg plays a big role there, I believe. But, the field could probably still exist without dg, I don't know too much about it. | |
| Sep 2, 2021 at 15:55 | comment | added | user347489 | @DanielWaters there you go! Gauge theory definitely came to mind, but I wasn't sure how heavily it's used in the standard model and such. Mechanics is another great example! | |
| Sep 2, 2021 at 15:53 | comment | added | Daniel Waters | @user347489 Gauge theory probably wouldn't exist, things like Hamiltonian mechanics probably wouldn't either (at least not to the degree of sophistication we have now). | |
| Sep 2, 2021 at 15:49 | comment | added | user347489 | Michael Engelhardt's answer is on point. Without DG we wouldn't have the theory of general relativity. My guess is that quantum field theory would also be heavily impaired without it. | |
| Sep 2, 2021 at 15:35 | answer | added | Daniel Waters | timeline score: 2 | |
| Sep 2, 2021 at 15:18 | comment | added | Ben McKay | This question will probably be closed as being unfocused. Please look at the explanations on this website about forms, connections, and so on, to see if some parts of your question are already answered. | |
| Sep 2, 2021 at 15:00 | review | Close votes | |||
| Sep 3, 2021 at 6:21 | |||||
| Sep 2, 2021 at 14:56 | comment | added | Michael Engelhardt | We want to understand black holes. | |
| Sep 2, 2021 at 14:48 | comment | added | Francesco Polizzi | Just to provide an example, the theorem of invariance of dimension is much easier to prove in the differentiable case, since it boils down to the standard fact that an isomorphism of finite-dimensional vector spaces preserves the dimension. If you want to prove it in the setting of topological manifold, where you do not have the tangent space, you need Brouwer's Invariance of Domain, that is a quite hard result. | |
| S Sep 2, 2021 at 14:36 | review | First questions | |||
| Sep 2, 2021 at 15:18 | |||||
| S Sep 2, 2021 at 14:36 | history | asked | Canrif | CC BY-SA 4.0 |