Timeline for What's the point of differential geometry? [closed]
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jun 14, 2023 at 6:22 | comment | added | Marcos Martínez Wagner | For me it all made scence when I learned about Riemannian Geometry. You want to do all the geometry Gauss made with surface, but "from the inside", living in a manifold that, a priori, is not embedd in the euclidean space (but actually it is because of Whitney theorem). Examples I like are space-time, proyective/grassmanian/flag manifold and the Tangent bundle. A Riemannian metric in this manifolds gives them actual geometry, we can talk about geodescis, volume, distance and many more things. We need differential geometry to define "Riemannian metric" (but is an area of reaserch of its own). | |
| Jul 18, 2021 at 20:18 | review | Reopen votes | |||
| Jul 20, 2021 at 12:58 | |||||
| Jul 18, 2021 at 18:42 | vote | accept | Daniel Waters | ||
| Jul 18, 2021 at 18:39 | history | closed |
Andrej Bauer LSpice godelian Wojowu Moishe Kohan |
Not suitable for this site | |
| Jul 18, 2021 at 17:39 | comment | added | user44143 | If you like unsolved problems, this answer by @alvarezpaiva is a good place to start: mathoverflow.net/a/137043/44143 | |
| Jul 18, 2021 at 17:19 | comment | added | Will Sawin | Some "big themes" I have noticed in differential geometry, as a non-expert, beyond what Ben McKay said, are "How can we define, and find, the `best' metric on a given manifold?" and "How can we find invariants of a Riemannian manifold that tell us something interesting about the dynamics of a classical or quantum particle on the manifold"? | |
| Jul 18, 2021 at 17:11 | answer | added | Lars | timeline score: 16 | |
| Jul 18, 2021 at 17:09 | comment | added | Branimir Ćaćić | I strongly second Ben McKay’s recommendation. A little basic classical mechanics will go a very long way towards making sense of the basic definitions of differential (especially Riemannian) geometry. I’d also recommend looking at the classical differential geometry of curves and surfaces (as culminating, say, with the Theorema Egregium and Gauß–Bonnet theorem)—a lot of basic differential geometry involves the intrinsic generalisation of conceptually transparent (but superficially extrinsic) constructions on submanifolds (e.g., curves and surfaces) of Euclidean space. | |
| Jul 18, 2021 at 17:06 | comment | added | Daniel Waters | @BenMcKay Thank you for your comment. Perhaps I have spent so long being lost in the details and the tools of differential geometry to really get the bigger picture. | |
| Jul 18, 2021 at 16:50 | comment | added | Ben McKay | The motivation of differential topology is to find invariants of manifolds under diffeomorphism, natural since the tools of calculus and differential equations use derivatives and not just continuity. But then Riemannian metrics provide a means of rigidifying (one of many means) which allows us to use analytic methods to relate global problems in differential topology to local curvature estimates. Similarly Lie groups allow us to use symmetry methods to study differential topology. So we look not just to differential topology but to a broad field of differential geometry. | |
| Jul 18, 2021 at 16:41 | review | Close votes | |||
| Jul 18, 2021 at 18:40 | |||||
| Jul 18, 2021 at 16:36 | comment | added | Daniel Waters | @WillSawin I edited the question. | |
| Jul 18, 2021 at 16:36 | history | edited | Daniel Waters | CC BY-SA 4.0 |
added 509 characters in body
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| Jul 18, 2021 at 16:32 | comment | added | Will Sawin | I think you will need to ask a more specific question to get a good answer on this site. At the very least you need to say what kinds of things you find motivating, e.g. applications to other branches of science, applications to other subfields of mathematics, beautiful structures, grand unsolved questions, etc. | |
| Jul 18, 2021 at 16:31 | comment | added | Ben McKay | You might want to read something about its applications in general relativity (perhaps from Weinberg) or classical mechanics (from Arnold). Reading from Lee gives a good introduction to the subject from the inside, but not to its applications. | |
| Jul 18, 2021 at 16:20 | history | asked | Daniel Waters | CC BY-SA 4.0 |