Timeline for Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$
Current License: CC BY-SA 3.0
20 events
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| Dec 4, 2016 at 6:56 | answer | added | Richard Montgomery | timeline score: 4 | |
| S Nov 11, 2016 at 13:01 | history | bounty ended | CommunityBot | ||
| S Nov 11, 2016 at 13:01 | history | notice removed | CommunityBot | ||
| Nov 4, 2016 at 15:47 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Nov 4, 2016 at 15:34 | comment | added | Ali Taghavi | @RyanBudney Do you mean $P(x,V)=exp(V/\sqrt{1+\parallel V \parallel^2}$? If you mean this equality, I think some thing is missing, for example it is not the case for $n=1$. | |
| Nov 4, 2016 at 14:22 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| S Nov 3, 2016 at 11:24 | history | bounty started | Ali Taghavi | ||
| S Nov 3, 2016 at 11:24 | history | notice added | Ali Taghavi | Draw attention | |
| Nov 1, 2016 at 20:22 | history | edited | Ali Taghavi |
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| Nov 1, 2016 at 19:22 | comment | added | Ali Taghavi | @RyanBudney But why does this imply that $X_{H}$ has the same dynamic as the geodesic flow?Moreover, what about the third part of my question? | |
| Nov 1, 2016 at 16:37 | comment | added | Ryan Budney | $P$ is pretty much precisely the exponential map if you re-scale the input vector by a non-linear function of its length. i.e. re-scale the input vector's length by $L \longmapsto 1/\sqrt{1+L^2}$. Exponentiate that vector, this gives you $P$. So your function $H$ is a similar non-linear rescaling of the length squared of the input vector. | |
| Nov 1, 2016 at 15:34 | comment | added | Ali Taghavi | @RyanBudney Do you mean that $X_{H}$ equal geodesic flow vector field up to a constant multiplier? Do you mean that $H$ is globally equal to $\parallel v \parallel ^2$ up to a constant?How can (3) be answered immediately? | |
| Nov 1, 2016 at 10:51 | comment | added | Ali Taghavi | @RyanBudney $H$ is not a re-scaled length of $v$, since the lenght of $v$ is an unbounded function but $H$ is a bounded function! Could you please elaborate your comment. I think your comments is not clear. | |
| Oct 31, 2016 at 23:33 | review | Close votes | |||
| Nov 1, 2016 at 9:55 | |||||
| Oct 31, 2016 at 23:17 | comment | added | Ryan Budney | $P$ is (up to a mild re-parametrization) the restriction of the exponential map for the tangent bundle of $S^n$. So 1, yes. $H$ is basically a re-scaled length of $v$, that answers (2). (3) can similarly be answered. | |
| Oct 31, 2016 at 21:07 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Oct 31, 2016 at 21:01 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Oct 31, 2016 at 19:46 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 31, 2016 at 19:41 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 31, 2016 at 19:32 | history | asked | Ali Taghavi | CC BY-SA 3.0 |