Timeline for Prescribing a Riemannian metric along a given geodesic
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 11, 2010 at 7:48 | comment | added | Sergei Ivanov | What regularity do you want from $g$? If $g$ is $C^2$, the geodesics must be $C^3$. | |
| Apr 11, 2010 at 1:02 | comment | added | Deane Yang | I don't see any issues with constructing the metric in the second case, except that everything has to match up properly at the endpoints. | |
| Apr 11, 2010 at 0:25 | comment | added | macbeth | For the second question, do you again have some $p$-jet condition? If not, can you construct the metric you want just by diffeomorph-ing a connected open neighbourhood of the two curves onto an open subset of a suitably-metrized sphere? (There will be a few cases depending on whether the angles of the curves at their meeting-points are acute-acute, acute-straight, acute-obtuse, etc.) | |
| Apr 10, 2010 at 23:47 | comment | added | Deane Yang | 1) Are you constructing the metric $g$ as $p$ times the Euclidean metric? 2) If you define $p$ along $\gamma$ so that $\gamma$ is a geodesic and has length 1, what else is there to do? | |
| Apr 10, 2010 at 22:38 | history | edited | Tom LaGatta | CC BY-SA 2.5 |
added 123 characters in body
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| Apr 10, 2010 at 8:35 | comment | added | Sergei Ivanov | If you prescribe the first derivatives of $g$ at $\gamma(0)$, the geodesic equation gives you a specific expression of $\gamma''(0)$ in terms of $\gamma'(0)$. So you cannot make any given curve a geodesic. | |
| Apr 10, 2010 at 6:53 | history | asked | Tom LaGatta | CC BY-SA 2.5 |