Timeline for Holonomic splitting
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2016 at 11:16 | vote | accept | Ben | ||
| Feb 10, 2016 at 10:46 | comment | added | Ben | Yes I see, I guess I have some other version :) | |
| Feb 10, 2016 at 10:45 | comment | added | Helene Sigloch | In the book I use (AMS, printed 2002), the section "Holonomic splitting" starts at the bottom of page 12 and ends at the top of page 14. But never mind, in my version of the book, the holonomic trivialisation is defined right above the theorem. | |
| Feb 10, 2016 at 10:40 | comment | added | Ben | You mean page 12 ? page 13 is the next chapter :). What exactly do you mean by "parallel" ? | |
| Feb 10, 2016 at 10:38 | comment | added | Helene Sigloch | By "horizontal sections from $\nu$" I mean sections "parallel" to the one we started with as defined on the top of page 13. | |
| Feb 10, 2016 at 10:37 | comment | added | Helene Sigloch | OK, it is confusing. You have sections of the original fibration $p: X \rightarrow V$, sections of the jet bundle and holonomic sections of the jet bundle, corresponding to sections of $p$ that satisfy the PDR. These are the ones we care about. By the holonomic splitting theorem, we know that once there is one local holonomic section, then you can "shift" this section parallelly inside a tubular neighbourhood and the parallel shifts will again be holonomic. But maybe I still didn't really understand your question. | |
| Feb 10, 2016 at 10:36 | comment | added | Ben | What do you mean by horizontal sections from $\nu$? | |
| Feb 10, 2016 at 10:26 | comment | added | Helene Sigloch | I am not sure about whether $\nu$ is diffeomorphic to $J^r( \mathbb{R}^n, \mathbb{R}^q)$. At least, that is not the point. The point is that the horizontal sections from $\nu$ are holonomic. I understand though that you are confused by the use of the term "section". Still, I think it makes no difference which point of view you take. | |
| Feb 10, 2016 at 10:21 | comment | added | Helene Sigloch | For the first two lines of your comment: Yes. The think the fibration property should be part of the definition of "tubular neighbourhood"? | |
| Feb 10, 2016 at 10:16 | comment | added | Ben | Is it correct what I have said above? If yes how can one prove such a thing? | |
| Feb 10, 2016 at 10:15 | comment | added | Ben | I do not understand how such a trivialisation looks like. Does it mean that if one picks a ball $U \subset V$ then one looks at the submanifold $F(U)\subset X^{(r)}$ and this submanifold has a tubular neighbourhood $\nu \subset X^{(r)}$ which at the same time is a fibration $\nu \rightarrow U$ such that $\nu$ is diffeomorphic to $J^{r}(\mathbb{R}^{n},\mathbb{R}^{q})$? I am a bit confused since by the term "section" the authors mean both the section as a map and its image. | |
| Feb 10, 2016 at 8:39 | history | answered | Helene Sigloch | CC BY-SA 3.0 |