Timeline for Is the Meusnier's theorem true for geodesics torsion?
Current License: CC BY-SA 4.0
8 events
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| Nov 9, 2023 at 10:30 | comment | added | Robert Bryant | @Bumblebee: The 'missing information' in $\Psi$ is the mean curvature $H=\tfrac12(h_{11}+h_{22})$ and a choice of orientation of the surface. However, given $I$ and $\Psi$, one can recover the mean curvature $H$ up to a sign since, by the definition of $\Psi$ and the Gauss equation $H^2 = K - \det_I(\Psi)$, and the orientation of the surface can also be recovered up to a sign. Thus, there are analogs of the Gauss-Codazzi equations for the pair $(I,\Psi)$, but they are more algebraically complicated than for the pair $(I,I\!I)$. | |
| Nov 8, 2023 at 18:06 | comment | added | Bumblebee | Do you know whether there is an analog of Gauss–Codazzi equations for Cartan's third fundamental form. A prior there is no reason to believe this is the case. It appears that this third fundamental form together with the first fundamental form is not enough to recover the second fundamental form. I'm trying to understand why this happens and what are the missing information. | |
| Oct 2, 2022 at 15:19 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed some typos and added Cartan's definition of his third fundamental form
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| Oct 1, 2022 at 16:13 | vote | accept | Bumblebee | ||
| Oct 1, 2022 at 15:28 | history | edited | Robert Bryant | CC BY-SA 4.0 |
After checking a copy of Cartan's EDS, I edited my answer to give more precise references.
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| Oct 1, 2022 at 10:56 | comment | added | Robert Bryant | I know that an English translation was made some years ago. I saw it because I was asked to comment on it. (It was a preliminary version and had many typos and translation errors.) As far as I know, it has never been published. The French is not that hard to read, and, in fact, for your purposes, you don't need to read the first 6 Chapters, which constitute Part I, an introduction to exterior differential systems. Part II starts with Chapter 7, the first 2 sections of which is a straightforward introduction to the moving frame (assuming only that you are familiar with differential forms). | |
| Oct 1, 2022 at 10:16 | comment | added | Bumblebee | Thank you very much. Is there any English translation of this book? | |
| Oct 1, 2022 at 9:54 | history | answered | Robert Bryant | CC BY-SA 4.0 |