Timeline for When is a level set an immersed submanifold?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2023 at 11:30 | comment | added | user1234567890 | @PierrePC thank you too. | |
| Jan 2, 2023 at 11:30 | comment | added | user1234567890 | @RyanBudney thank you. | |
| Dec 30, 2022 at 17:20 | comment | added | Ryan Budney | For a Morse function the level sets are not immersed manifolds, in general. There is an onto map to the level sets, but its derivative has rank $1$ on a large subspace. So in the special case your ambient manifold has dimension $2$, this isn't a problem (as Pierre PC observed) but when your ambient manifold is $3$ dimensional (or higher dimensional) this is a real problem, in the smooth setting. If you are okay living in the topological world, yes you can resolve the problem for certain types of critical points, but not all. | |
| Dec 30, 2022 at 12:47 | comment | added | user1234567890 | @RyanBudney so if every critical point is Morse, level sets are locally smooth immersed submanifolds and globally topological immersed manifolds? Or level sets are globally topological immersed manifolds no matter what condition you put to critical points? | |
| Dec 25, 2022 at 21:21 | comment | added | Pierre PC | @RyanBudney Ah yes, I see. You're saying the pinching of a cylinder into a cone is not an immersion; I had a wrong picture in mind indeed. | |
| Dec 25, 2022 at 21:14 | comment | added | Ryan Budney | @PierrePC: Apologies, I grabbed too simple an example. Take for example, $x^2+y^2-z^2=0$. This is a purely local problem. | |
| Dec 25, 2022 at 21:11 | comment | added | Pierre PC | @RyanBudney I am not sure I understand your example. Are you thinking of the height function for a donut balancing on its tip (or say a tire in its normal use position)? In this situation I don't see the problem, since the singular cases are points or figure eights. | |
| Dec 25, 2022 at 21:00 | comment | added | Ryan Budney | @PierrePC: the level sets in that case are not immersed manifolds, in general. Locally yes it's true, but there is a global problem. Take for example the standard linear height function on a torus. You need to leave the smooth category and be okay with topological immersions to make sense of your answer. I don't think there will be a non-tautological answer that is satisfactory to the OP. The main reason is that immersed submanifolds aren't level sets of any natural family of functions on manifolds that I know of. | |
| Dec 25, 2022 at 11:53 | comment | added | user1234567890 | @PierrePC anyway thank you for your comment. Now I have somewhere to start. | |
| Dec 25, 2022 at 9:47 | comment | added | Pierre PC | I am fairly certain this will be true if every critical point for $c$ is Morse (i.e. for every $x$ such that $f-c$ vanishes at the second order, the Hessian is non degenerate) by blowing up the singularities at those points. This is a generic condition on $f$, so for most $f$ all the level sets will be immersed manifolds. That said, there must be someone more qualified for a complete answer on MO. | |
| Dec 25, 2022 at 9:03 | history | edited | user1234567890 | CC BY-SA 4.0 |
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| Aug 28, 2022 at 6:46 | history | edited | user1234567890 | CC BY-SA 4.0 |
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| Aug 28, 2022 at 6:46 | comment | added | user1234567890 | @RyanBudney true. I'll delete it. | |
| Aug 28, 2022 at 3:49 | comment | added | Ryan Budney | I might take an issue with your question. When you say you are asking for the "next level" of generality, what do you mean? Can you describe all the levels of generality in some system? I think usually people would take your line of questioning in a different direction, i.e. have a disagreement on the form of generalization. | |
| Aug 27, 2022 at 21:21 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing
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| Aug 27, 2022 at 19:09 | history | asked | user1234567890 | CC BY-SA 4.0 |