Timeline for Affine structure on the circle whose atlas consists of homeomorphisms onto $\mathbb{R}$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 23 at 13:48 | answer | added | user515519 | timeline score: 2 | |
| Jan 23 at 8:29 | comment | added | YCor | @RyanBudney For an exotic circle $\mathbf{R}_{>0}/\langle t\rangle$ holonomy yields locally multiplication by $t$, so you don't consistently get a Riemannian metric in this way. | |
| Jan 23 at 8:16 | comment | added | Sergiy Maksymenko | Thank you very much, Ryan. Your arguments probably extend to compact affine manifolds, so they might not have affine atlases with surjective charts, since otherwise they would have infinite volume in some metric constructed by that atlas | |
| Jan 23 at 8:00 | answer | added | YCor | timeline score: 6 | |
| Jan 23 at 7:42 | comment | added | Ryan Budney | One way to approach this (perhaps not the most efficient) would be to use the affine charts to put a Riemann metric on the manifold, by pulling back the standard metric on the co-domains. If your charts had co-domain $\mathbb R$ your circle would have infinite length, contradicting compactness. | |
| Jan 23 at 7:16 | history | edited | Sergiy Maksymenko | CC BY-SA 4.0 |
added 4 characters in body
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| Jan 23 at 6:57 | history | asked | Sergiy Maksymenko | CC BY-SA 4.0 |