Timeline for How to "globalize" the inverse function theorem?
Current License: CC BY-SA 2.5
11 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Nov 29, 2010 at 16:23 | comment | added | Deane Yang | If the Jacobian vanishes on a submanifold, then that's where all the action is. You have to study the behavior of $F$ on and near the singular submanifold in order to say anything more. There is not going to be a general theorem about this, since the behavior can be quite complicated. But perhaps your situation has some additional conditions (dimensional constraints, for example) that might reduce the possibilities. | |
| Nov 29, 2010 at 5:51 | history | edited | anonymous | CC BY-SA 2.5 |
added 142 characters in body
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| Nov 29, 2010 at 5:41 | comment | added | anonymous | @Deane: Thanks for your input; however, in the applications I have in mind I must allow the Jacobian of $F$ to vanish at some points (or rather on a submanifold of $V\times W$ of nonzero codimension in $V\times W$), so I can't genuinely globalize $f$, that's the whole point. | |
| Nov 28, 2010 at 22:49 | answer | added | Andrey Rekalo | timeline score: 9 | |
| Nov 28, 2010 at 22:30 | history | edited | anonymous | CC BY-SA 2.5 |
typo fixed
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| Nov 28, 2010 at 22:21 | comment | added | Deane Yang | Have you tried to work this out yourself? The inverse function theorem gives uniqueness as well as existence. What goes wrong when you try to extend the domain of the inverse map? | |
| Nov 28, 2010 at 22:15 | history | edited | anonymous | CC BY-SA 2.5 |
typo fixed
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| Nov 28, 2010 at 21:59 | history | edited | anonymous | CC BY-SA 2.5 |
another typo fixed
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| Nov 28, 2010 at 21:39 | history | edited | anonymous | CC BY-SA 2.5 |
some typos fixed and notation slightly changed
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| Nov 28, 2010 at 21:08 | history | asked | anonymous | CC BY-SA 2.5 |