Timeline for Palais's and Kobayashi's theorems on automorphism groups of geometric structures
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 21, 2022 at 14:24 | comment | added | Chris Wendl | I thought some more, and now I think I know what you were getting at with foliations, but I still perceive a problem. It is true that in Kobayashi's setting, the orbit is a union of leaves of a foliation. If I knew that it contains only countably many leaves, I could argue via Baire that being closed implies it is a submanifold. But this is why I asked Question 1: the orbit might really have uncountably many components, in which case Baire does not seem applicable. The more I think about it, the more this approach to the proof seems like a dead end, but I'd love to be proven wrong. | |
| Mar 21, 2022 at 13:14 | comment | added | Chris Wendl | I only just now noticed that you added the statement "if an orbit is closed, then it is an embedded submanifold" to your answer. Does that follow from something in the book? | |
| Mar 20, 2022 at 15:37 | comment | added | Chris Wendl | OK, point taken that orbits looking the same everywhere rules out my horror scenario, though I don't see an obvious way from there to the conclusion that no other horror scenarios are possible. Can you say more specifically what properties of foliations you are referring to? About the connected component of the orbit: in theory yes, but if I don't yet know that it's a submanifold, then I don't know whether "component" and "path-component" (or also "smooth path-component", which is what you use in defining initial submanifolds) are the same thing. That is the tricky part. | |
| Mar 19, 2022 at 8:09 | comment | added | Peter Michor | An orbit looks the same everywhere, so your horror scenario cannot happen: the orbit consisting of disjoint submanifolds would fill densely their closure, which is of higher dimension. See the description of foliations (with jumping dimensions) in the source I gave. Moreover: Are you not interested only in the connected component of the orbit. The answer to your first question should follow from the fact, that the group keeps invariant a geometric structure, and should involve the type of structure. | |
| Mar 18, 2022 at 9:30 | comment | added | Chris Wendl | How does the orbit being closed solve the topology question? The horror scenario I described in my question (a converging sequence of disjoint (n-1)-submanifolds) is also a closed initial submanifold, as far as I can see. | |
| Mar 18, 2022 at 8:16 | history | edited | Peter Michor | CC BY-SA 4.0 |
added 74 characters in body
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| Mar 18, 2022 at 8:06 | comment | added | Peter Michor | The point is: An orbit of a Lie group action is better than just an immersed submanifold. The toplogy question is solved since the orbit is closed. | |
| Mar 17, 2022 at 7:41 | comment | added | Chris Wendl | Doesn't inserting the word "initial" basically remove the difficulty by redefining the goal? The neighborhood of a point in an initial submanifold can still look much uglier than an actual submanifold, no? I don't want to have to change the topology on the orbit before saying that it has a nice structure. | |
| Mar 16, 2022 at 19:24 | history | answered | Peter Michor | CC BY-SA 4.0 |