Timeline for Variation of curvature with respect to immersion?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 8, 2015 at 18:57 | comment | added | Omega Tree | Thank you very much, Robert! I finally had the chance to work through it all, and have learned a great deal in the process. For me, the key piece was adding the $\mu dt$ term to the Darboux framing assumption. This setup is proving quite useful in my research. Thanks again! | |
| Dec 7, 2014 at 15:06 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Added a bit more detail in the calculation
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| Dec 6, 2014 at 23:17 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Added the derivation of the variational formula, as requested
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| Dec 4, 2014 at 22:17 | comment | added | Omega Tree | It seems like the trick may be to use a Darboux frame? There you get $\omega_{31}=\kappa_1\omega_1$. But if I then differentiate in $t$ I get stuck, because I do not know how the principal directions change. (Apologies if I am missing something obvious - I am new to this whole business of moving frames!) | |
| Nov 2, 2014 at 17:25 | comment | added | Omega Tree | Thanks Robert - no rush, just a friendly bump. :-) | |
| Nov 1, 2014 at 13:43 | comment | added | Robert Bryant | @OmegaTree: Sorry, I've been really busy lately and haven't had time to put this in. I'll try to get to it this weekend. | |
| Nov 1, 2014 at 13:31 | comment | added | Omega Tree | Hi Robert, any chance you might still say something about how to set up moving frames on an evolving surface? Sorry to be a pest, but it appears to be a very useful tool! | |
| Oct 20, 2014 at 18:41 | comment | added | Omega Tree | Thanks Robert. Whenever you do get a spare moment, I think it would be very helpful to have such a description archived with the results above. | |
| Oct 18, 2014 at 13:36 | comment | added | Robert Bryant | @OmegaTree You're welcome. Sure, I'll write a description of the calculation, but I probably won't have time to do it until this evening (I'm on the East Coast of the US). | |
| Oct 18, 2014 at 12:56 | comment | added | Omega Tree | Thanks Robert. I'm accepting this answer because it gives an explicit expression for the variations, but of course the other answers below are also useful. However, my experience with moving frames is limited and I was unable to re-derive the result myself. The main difficulty is that I do not know how to set up a moving frame on an evolving surface in an intelligent way, so that I don't run into the same difficulties as with other methods of derivation. Can you share at least the ansatz you used for this calculation? Thanks! | |
| Oct 18, 2014 at 12:54 | vote | accept | Omega Tree | ||
| Oct 8, 2014 at 1:05 | history | edited | Robert Bryant | CC BY-SA 3.0 |
clarified the $\delta$-terminology
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| Oct 7, 2014 at 17:55 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added missing factors of 1/2 and 2 in the formula for the variation of H and K
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| Oct 7, 2014 at 15:14 | history | answered | Robert Bryant | CC BY-SA 3.0 |