Timeline for Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis?
Current License: CC BY-SA 4.0
11 events
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| Dec 21, 2020 at 22:30 | comment | added | user44143 | @PaulSiegel, the OP used the word “intrinsic”, and that’s not the way I would describe an approach having key definitions based on comparisons with Euclidean space. But I agree that the contemporary research like Busemann’s is closer to the Burago-Burago-Ivanov approach. | |
| Dec 21, 2020 at 16:12 | comment | added | Matko | Thaugh these books and perelman all used coordinates in some way, and that's probably the best way to achieve what they intended, | |
| Dec 21, 2020 at 16:08 | comment | added | Matko | Exactly, it is all about the primitives of distances lenghts and angles, and I find this in the long run to be very practical, as supported by perelman and his proof, I believe through even more painful refinement, even more great results would come simply by reducing the complexity of the underlying machinery | |
| Dec 21, 2020 at 16:00 | comment | added | Paul Siegel | Other good stuff to read includes the papers of both Buragos, Perelman, Ivanov, and Petrunin - the latter two are active on MO from time to time. | |
| Dec 21, 2020 at 15:58 | comment | added | Paul Siegel | There has been a lot of progress since that book was written - you might consider the more recent textbook "A Course on Metric Geometry" by Burago, Burago, and Ivanov. One of the premises of this area is that if you can prove some of the main theorems of Riemannian geometry using just low-level concepts like distances between points, lengths of curves, angles between curves, etc. then you can generalize them to spaces with singularities - this was one of the key perspectives that Perelman brought to the Poincare conjecture, for example. | |
| Dec 21, 2020 at 15:57 | comment | added | Deane Yang | @ArcDDD, Henri's father Elie used frames. Henri worked on more modern coordinate-free formulations of aspects of differential geometry. | |
| Dec 21, 2020 at 15:09 | comment | added | Matko | I don't have much access to his work, but he used frames which is pretty much the same thing | |
| Dec 21, 2020 at 15:07 | vote | accept | Matko | ||
| Dec 21, 2020 at 14:18 | comment | added | Deane Yang | Henri Cartan and others did find ways to do differential forms without coordinates. But these approaches usually obscure rather than elucidate what's going on. | |
| Dec 21, 2020 at 14:09 | comment | added | Matko | This description matches exactly to what I had in mind, I always considered geodesics more fundamental than covariant derivative, dif forms i find quite esthetically pleasing as a concept but couldnt find a way where they are not irreducible to coordinates | |
| Dec 21, 2020 at 13:50 | history | answered | user44143 | CC BY-SA 4.0 |