Timeline for First order estimates of geodesic normal coordinates
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2020 at 19:45 | vote | accept | Totoro | ||
| Oct 28, 2020 at 14:23 | answer | added | Robert Bryant | timeline score: 9 | |
| Oct 28, 2020 at 3:02 | comment | added | user44143 | GabeK and DeaneYang, apparently one of you has not upvoted this question -- but the discussion suggests that you do find it surprisingly difficult, so probably worth an upvote. | |
| Oct 28, 2020 at 0:41 | comment | added | Deane Yang | Actually, what I wrote in my last comment isn't quite right either. What is true is that, to get a bound on the angular derivatives of the metric using only the Jacobi equation, you need a bound on the covariant derivative of the curvature. However, there are additional equations coming from the angular components of the curvature tensor. Perhaps these could be used somehow to get the desired bound. I don't think this is known, which is also why I'm not sure whether any counterexample is known or not. | |
| Oct 27, 2020 at 19:47 | comment | added | Totoro | @DeaneYang You are right. I was wondering if there is an explicit counterexample. | |
| Oct 27, 2020 at 18:24 | comment | added | Deane Yang | Let me add another comment about this: Exponential coordinates satisfy only an ODE, namely the Jacobi equation. So bounded curvature gives you a bound, in terms of the curvature, for only in the radial direction. To get a bound on the angular derivatives of the metric, you need to use the derivative of the Jacobi equation in the angular directions, so the covariant derivative of curvature appears. | |
| Oct 27, 2020 at 15:13 | history | edited | Totoro | CC BY-SA 4.0 |
added 219 characters in body
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| Oct 27, 2020 at 15:01 | comment | added | Gabe K | I deleted my answer. The comments by Deane Yang and Totoro were right and I'm not sure how to bound the non-radial directions. | |
| Oct 27, 2020 at 14:51 | comment | added | Deane Yang | I've deleted my original comment. My first statement, that a sufficiently thin flat cylinder is a counterexample, was correct. However, offhand, I don't see how to get the bound using exponential coordinates. You can get the $C^1$ bound on the metric using either harmonic or almost linear coordinates (as defined by Jost and Karcher). | |
| S Oct 27, 2020 at 14:44 | history | suggested | gmvh |
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| Oct 27, 2020 at 14:13 | review | Suggested edits | |||
| S Oct 27, 2020 at 14:44 | |||||
| Oct 27, 2020 at 14:03 | history | asked | Totoro | CC BY-SA 4.0 |