Timeline for The distance to the zero section of $TM$
Current License: CC BY-SA 4.0
29 events
| when toggle format | what | by | license | comment | |
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| Jan 24, 2023 at 18:35 | vote | accept | Ali Taghavi | ||
| Jan 8, 2023 at 22:43 | answer | added | Anton Petrunin | timeline score: 2 | |
| Dec 19, 2020 at 19:19 | comment | added | Gabe K | By choosing a pair of connection and metric, you can use this construction to find complex/ almost-complex manifolds with very interesting geometry. It's also worth noting that the answer to your original question is negative if you use this more general construction. | |
| Dec 19, 2020 at 19:14 | comment | added | Gabe K | You can use an arbitrary affine connection (independent of the metric) to define the splitting and this is still called a 'Sasaki metric.' A good reference on the topic is the following paper by Satoh: arxiv.org/abs/1908.10824 | |
| Dec 19, 2020 at 17:26 | comment | added | Ali Taghavi | @GabeK your comments leads me to the following question:"Are there some other kind of Sasaki metric'? | |
| Dec 18, 2020 at 20:34 | comment | added | Ali Taghavi | @RobertBryant BTW For what kind of principal bundles this pathology would not occure? that is for what kind kind of principal bundles the geodesics joining two points on the same fibers must be vertical curves? | |
| Dec 18, 2020 at 20:27 | comment | added | Ali Taghavi | @RobertBryant so we can construct compact Riemanian manifold $M,N$ with Riemannian submersion such that the minimizing curve joining two cofiber points is not necessarily a vertical curve:We choose a huge $2$-torus $\mathbb{T}^2$ and the same action you mentioned on $\mathbb{R}\times \mathbb{T}^2 $, instead of $\mathbb{R}\times \mathbb{R}^2$. | |
| Dec 18, 2020 at 11:02 | comment | added | Robert Bryant | @AliTaghavi: As for the main question, Petrunin's argument is correct. | |
| Dec 18, 2020 at 1:45 | comment | added | Ali Taghavi | @RobertBryant Thank you. In fact you consider the direct sum of the Mobius bundle with itself. BTW what can be said about the main question of this post? | |
| Dec 18, 2020 at 1:15 | comment | added | Robert Bryant | @AliTaghavi: Yes, there is an orientable counterexample. Just let $M=\mathbb{R}^3/\mathbb{Z}$ with action $n\cdot(x,y,z) = (x{+}2\pi\,,\,(-1)^ny\,,\,(-1)^nz\,)$ and let the submersion $\sigma:M\to S^1$ be $\sigma([x,y,z])=\mathrm{e}^{ix}$. The same construction works with $p=[0,a,0]$ and $q = [0,-a,0]=[2\pi,a,0]$, but now $M$ is orientable. | |
| Dec 17, 2020 at 22:52 | comment | added | Ali Taghavi | @RobertBryant Is there a counter example with extra condition of orientability? | |
| Dec 15, 2020 at 15:53 | comment | added | Ali Taghavi | @RobertBryant Very interesting example! | |
| Dec 15, 2020 at 15:27 | comment | added | Ali Taghavi | @RobertBryant Thank you very much for your attention to my question and your comment. I realize from your comment that the total space of the Mobius bundle is a counter example that is a geodesic joining two cofiber points are not necessarilly vertical. | |
| Dec 15, 2020 at 14:02 | comment | added | Robert Bryant | @AliTaghavi: In response to your question about the more general case of a Riemannian submersion (and where I assume that you meant $p=v$ and $q=w$), the answer is a definite 'no', even in the case that the source and target of the Riemannian submersion are complete. Just take the standard flat metric on $M = \mathbb{R}^2/\mathbb{Z}$ where the action is $n\cdot(x,y) = (\,x+2\pi\,n,\, (-1)^ny\,)$ and the submersion is $\sigma([x,y]) = \mathrm{e}^{ix}$ and look at the shortest path from $[0,a]$ to $[0,-a] = [2\pi, a]$ for $a>\pi$. | |
| Dec 11, 2020 at 21:52 | history | undeleted | Ali Taghavi | ||
| Dec 11, 2020 at 18:57 | history | deleted | Ali Taghavi | via Vote | |
| Dec 10, 2020 at 21:34 | history | edited | Ali Taghavi |
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| Dec 10, 2020 at 12:26 | comment | added | Ali Taghavi | @AntonPetrunin thank you for your comment. The gradient of length function is veryical then are you showing the geodesic paths? Is TM complet? I have another related question: let we have a Riemannian submersion and v, w liy on the same fiber. Does shortest path from p to q is necessarily a geodesic in the same fiber?(we do not assume any completeness). BTW may you ellaborate your answer? | |
| Dec 10, 2020 at 11:41 | comment | added | Ali Taghavi | @GabeK yes I mean the metric on TM arising from $\nabla$_ isomorphism between horisontal space and $T_p M$ from one hand and the vertical space and $T_p M$ from another hand. | |
| Dec 10, 2020 at 11:38 | comment | added | Ali Taghavi | @LSpice thank you for your edit! | |
| Dec 10, 2020 at 5:44 | comment | added | Anton Petrunin | Yes. Consider the function $f\colon V\mapsto |V|$. Note that $\nabla_Vf$ has vertical part $V/|V|$ if $V\ne0$ and its horizontal part vanishes. Make a conclusion. | |
| Dec 10, 2020 at 1:08 | comment | added | Gabe K | By Sasaki metric you mean the one using the Levi-Civita connection, correct? | |
| Dec 9, 2020 at 23:59 | comment | added | LSpice | You had $V_p \in TM$, which was surely meant to be $V_p \in T_pM$. I changed it. | |
| Dec 9, 2020 at 23:58 | history | edited | LSpice | CC BY-SA 4.0 |
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| Dec 9, 2020 at 23:45 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Dec 9, 2020 at 23:39 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Dec 9, 2020 at 23:26 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Dec 9, 2020 at 23:20 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Dec 9, 2020 at 23:11 | history | asked | Ali Taghavi | CC BY-SA 4.0 |