Timeline for Cotangent bundle lift theorem
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2022 at 1:31 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
a minor typo
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| Jan 16, 2022 at 19:08 | comment | added | JHM | @glS You are correct I wrote composition in the wrong order, thank you for clarifying! I was trying to say $d\pi \circ d(f^*)=df \circ d\pi$, as per the commutative diagram $f \circ \pi = \pi \circ f^*$. But rereading my answer, I dont think I fully understood the question when I wrote it. | |
| Jan 16, 2022 at 19:04 | history | edited | JHM | CC BY-SA 4.0 |
deleted 16 characters in body
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| Jan 16, 2022 at 18:49 | comment | added | glS | isn't $d(f^*)\circ d\pi$ ill-defined here? We have $\pi:T^* M\to M$, thus $d\pi:TT^* M\to TM$, but $d(f^*): TT^*M\to TT^* M$. So $d(f^*)$ takes as input elements of $TT^*M$ while $d\pi$ outputs elements of $TM$. Same for $d\pi\circ df$: $df$ gives things in $TM$ wihle $d\pi$ wants things in $TT^* M$. Am I misunderstanding something? The relation makes more sense if written as $d\pi\circ d(f^*)=df^{-1}\circ d\pi$, I think, which should follow from $f^{-1}\circ \pi=\pi\circ f^*$, due to $f^*(\alpha_p)\equiv\alpha_p\circ df|_{f^{-1}(p)}\in T^*_{f^{-1}(p)}M$ if $\alpha_p\in T_p^*M$. | |
| Dec 8, 2013 at 0:47 | vote | accept | Pengfei | ||
| Dec 8, 2013 at 0:47 | comment | added | Pengfei | I checked the reference provided by alvarezpaiva. By considering the symplectic vector field induced by $\theta$, such $F$ must preserve the fibers. The pullback relation gives $\pi\circ F=f^{-1}\circ \pi$, which is used to prove $f^\ast\theta=\theta$. There is another typo at the end of the second paragraph: $\pi\circ G=\pi$. Now I understand your answer. Thank you! | |
| S Dec 7, 2013 at 19:42 | history | suggested | hoj201 | CC BY-SA 3.0 |
If $G:T^*M \to T^*M$ is a fiber map over the identity when $\pi \circ G = \pi$. I think you wrote $dG \circ d\pi = d\pi$ when you meant to write $d\pi \circ dG = d\pi$. Also a parenthesis was missing.
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| Dec 7, 2013 at 18:48 | review | Suggested edits | |||
| S Dec 7, 2013 at 19:42 | |||||
| Dec 7, 2013 at 5:01 | history | edited | JHM | CC BY-SA 3.0 |
minor re-wording
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| Dec 7, 2013 at 4:54 | history | edited | JHM | CC BY-SA 3.0 |
minor wording
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| Dec 7, 2013 at 4:33 | history | undeleted | JHM | ||
| Dec 7, 2013 at 4:00 | history | deleted | JHM | via Vote | |
| Dec 7, 2013 at 3:14 | history | answered | JHM | CC BY-SA 3.0 |