Timeline for Extrinsic horizontal path lifting
Current License: CC BY-SA 4.0
18 events
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| Sep 23, 2023 at 14:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
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| May 11, 2020 at 5:29 | answer | added | PPR | timeline score: 1 | |
| May 8, 2020 at 10:21 | comment | added | S.Surace | Maybe the chapter 'rolling without twisting or slipping' in Appendix B of Sharpe's Differential Geometry book might be an interesting read, it sounds similar to what you are trying to do, and can be applied extrinsically. | |
| May 6, 2020 at 15:32 | comment | added | PPR | @IgorBelegradek, as far as I understood from the construction of the orthonormal frame bundle, there is a 1-1 correspondence between paths in $\mathbb{R}^2$ and paths in $\mathbb{S}^2$ in an isometric way, which doesn't require charts. The question is how to realize this 1-1 map explicitly, thinking extrinsically of the sphere embedded in $\mathbb{R}^3$, starting from a curve in $\mathbb{R}^2$. In your comment you fix two curves on the two spaces, but it's not clear (to me) how to relate the two | |
| May 6, 2020 at 15:21 | comment | added | Igor Belegradek | You need to edit the question so that it makes sense mathematically. Otherwise, it may get closed as "unclear what you are asking". It seems this is what you want: fix a unit speed parametrization of a curve in $\mathbb S^2$ and $\mathbb R^2$. Then any other parametrization differs from the fixed one by a self-map $\phi$, and then the new speed will be $|\phi^\prime|$. This is true both in $\mathbb S^2$ and $\mathbb R^2$. Am I missing something? | |
| May 6, 2020 at 14:56 | comment | added | PPR | @IgorBelegradek, thanks for the comment. I understand this depends on the parametrization, but I exactly want to capture that (i.e. not allow for any parametrization), but stick to a given one that whomever gave me $w$ chose. When I say lift (technically) I mean a lift of curves from the sphere to the orthonormal frame bundle. I think my usage of the word 'lift' to denote taking a curve in Euclidean space to the sphere was technically incorrect. | |
| May 6, 2020 at 14:46 | comment | added | Igor Belegradek | I haven't read the whole question but to talk about lifting you need a map $\mathbb S^2\to \mathbb R^2$. Which map are you using? Without a map the word "lift" has no meaning. In general if a map is a Riemannian submersion, then you can lift any path in the base to a horizontal path in the total space. Any Riemannian submersion of manifolds of the same dimension is a local isometry. Round sphere is not locally isometric to the Euclidean plane. What you call "energy" is actually the square of the speed. It depends on how you parametrize the curve. Any curve can be parametrize by unit speed. | |
| May 6, 2020 at 14:26 | history | asked | PPR | CC BY-SA 4.0 |