Timeline for Continuity of the 1-jet prolongation map $\text{imm}(M,N)\to \text{fimm}(M,N)$
Current License: CC BY-SA 4.0
9 events
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| Jul 14, 2023 at 23:56 | comment | added | heervande | Reading my above comment again, I see that I am still treating this comment section as a 'forum for grad student homework problems'. Nonetheless, I'd still like to know the answer but I get it if this is not for this site. | |
| Jul 14, 2023 at 19:52 | comment | added | heervande | Yes, I understand that. The reason I specified the topology on $\text{fimm}(M,N)$ is because of your remark that this map is continuous for $k\geq 1$. Is it not continuous for $k=0$ then? In other words, I'm not asking for a solution but rather whether if I have the spaces defined correctly like this. | |
| Jul 14, 2023 at 16:49 | comment | added | Ryan Budney | Sure, but this isn't a forum for grad student homework problems. I'm just trying to give you a sense for a first approach. Once you see a way to prove a theorem in a special case it's often fairly easy to extrapolate to full-generality. | |
| Jul 14, 2023 at 10:13 | comment | added | heervande | @RyanBudney I have already taken a look at Hirsch's book, and what I get from the article, Weiss uses compact-open (what Hirsch calls weak) $C^\infty$ topology on the space of immersions, whereas the topology on the space of formal immersions results from the subspace of $C^0(M,N)\times C^0(TM,f^*TN)$ where both spaces in the product are equipped with the compact open (weak) C^0 topology. I don't believe we are allowed to assume $M$ compact in this case, so that means it matters if we work with the strong or weak topology, right? | |
| Jul 14, 2023 at 5:53 | comment | added | Ryan Budney | Perhaps that does not fully answer your question, but it should get you started. Do take a look at Hirsch's book on Differential Topology. | |
| Jul 14, 2023 at 5:52 | comment | added | Ryan Budney | If you use the $C^k$-topology on both spaces, it is continuous for all $k \geq 1$. Which definition of the $C^k$-topology are you using? There are a variety of equivalent definitions. Hirsch's book is quite good about running through the equivalent definitions. For example, the simplest argument involves the $C^1$-topology which you could describe as the uniform topology when you restrict the derivative of an immersion to its unit tangent bundle. In that regard, your topology on your domain and codomain are given by essentially the same metric. This of course assumes $M$ compact. | |
| Jul 13, 2023 at 13:38 | history | edited | heervande | CC BY-SA 4.0 |
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| S Jul 13, 2023 at 13:37 | review | First questions | |||
| Jul 13, 2023 at 14:03 | |||||
| S Jul 13, 2023 at 13:37 | history | asked | heervande | CC BY-SA 4.0 |