Timeline for Second derivative of Riemannian Exponential Map
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| May 29, 2017 at 16:43 | comment | added | ABIM | Cool, I'll look it up. Feel free to add the ref whenever you have access to your books :) | |
| May 26, 2017 at 18:55 | comment | added | Richard Montgomery | @CSA, Matthias Ludewig. Apologies for not remembering the ref. I would start looking probably in Berger's Geometry in the 2nd half of the 20th century, or Cheeger-Ebin. (I'm not near access to books right now). | |
| May 25, 2017 at 16:35 | comment | added | ABIM | @RichardMontgomery yes I beleive I red this also but I can't find a reference. | |
| Mar 14, 2014 at 2:01 | comment | added | Matthias Ludewig | How do you prove that? | |
| Mar 13, 2014 at 6:01 | comment | added | Richard Montgomery | I don't know about this question but I did learn recently that the 4th jet of the function (s,t) -> d(exp(sv), exp(tw)) where the `exp' is taken at a fixed pt. p yields the Riem curv tensor R_p (v,w,v,w) | |
| Feb 23, 2014 at 8:23 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
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| Feb 23, 2014 at 7:16 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
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| Feb 23, 2014 at 6:24 | comment | added | Deane Yang | If derivative of the exponential map can be described in terms of Jacobi fields, then the second derivative should have a corresponding description in terms of the covariant derivatives of the Jacobi fields. And since a Jacobi field satisfies the Jacobi equation, its covariant derivative satisfies an equation obtained by taking the covariant derivative of both sides of the Jacobi equation. | |
| Feb 23, 2014 at 3:37 | comment | added | Matthias Ludewig | I do not really understand by what I should differentiate the Jacobi Equation. Could you elaborate? | |
| Feb 22, 2014 at 20:33 | comment | added | Deane Yang | Since the first derivative of the exponential map are Jacobi fields, it makes sense that the second derivative are derivatives of the Jacobi fields. And just as Jacobi fields satisfy a linear second order ODE along a geodesic ray, their derivatives satisfy the second order ODE that you get by differentiating the first ODE. | |
| Feb 22, 2014 at 19:55 | history | asked | Matthias Ludewig | CC BY-SA 3.0 |