Timeline for Finding covariant derivative of a riemanian submanifold
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jul 26, 2010 at 18:11 | comment | added | tamir | PS (no place left up there) thanks for the time and effort, Tamir. | |
| Jul 26, 2010 at 18:10 | comment | added | tamir | About the second approach you showed me, I see that $ P $ is doing the "magic" there. But does $ P $ have an "analytical" definition of some sort that I can work with? My problem started with the fact that when I tried to project $\nabla^N$ to $\nabla^M$ I took $\nabla^N_{\partial i}\partial j$ and I wrote it explicitly with $\partial k$ and "threw away" the term with $ k = m+1 $. This is the projection as I understand it. But the Christoffels in the other $\partial k, k \neq m+1$ stuck me. Because then, I didn't know what how to "project" them. Can you give me anopther word on that, please? | |
| Jul 26, 2010 at 18:10 | comment | added | tamir | In the first part, given a metric tensor, is it OK just to choose new local coordinates and preserve the tensor? I think that as long as I choose a smooth coordinate change, than maybe g takes another form, but it still operates the same on members of $ TN $ . It's like doing $ w (C^TgC) v = (Cw)^T * g * (Cv) $ right? (if it's not too much trouble - if it is so, since $ g $ is symmetric and strictly positive, so it can always be diagonalized and therefore we have sort of "principal" directions where the arc length is just $ ds^2 = Udu^2 + Vdv^2 + W*dw^2 + ... $ ?) | |
| Jul 26, 2010 at 18:10 | comment | added | tamir | Hi Greg, I'm still not sure about two points here, so let me see if I get it correctly. | |
| Jul 26, 2010 at 0:55 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
deleted 7 characters in body
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| Jul 26, 2010 at 0:29 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |