Timeline for Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?
Current License: CC BY-SA 3.0
16 events
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| Jan 6, 2020 at 18:58 | comment | added | Pedro Lauridsen Ribeiro | Conversely, given $P$ as defined in the question above, it is clear that $[P,f]=$ commutator of $P$ and (multiplication by) a smooth function $f$ on $M$ is a linear differential operator of order $k-1$. (Remark: one has to add to your definition the "initial condition" that linear differential operators of order $-1$ are identically zero) | |
| Jan 6, 2020 at 18:56 | comment | added | Pedro Lauridsen Ribeiro | @AliTaghavi Not really - such a definition is essentially the same as the one in terms of jet bundles. To see that, it suffices to know that the latter's fiber over $p\in M$ may be defined as the quotient of the space $E(p)$ of germs of smooth sections of $\pi$ over $p$ modulo the subspace of such germs vanishing to order $k+1$ at $p$. As such, it can be seen by induction on $k$ that given smooth functions $f_1,\ldots,f_{k+1}$ on $M$, one always has $D((f_1-f_1(p))\cdots(f_{k+1}-f_{k+1}(p))s)(p)=0$ for all $p\in M$, $s\in\Gamma(\pi)$, hence $D$ is necessarily of the form stated for $P$ above. | |
| Dec 28, 2019 at 18:26 | comment | added | Ali Taghavi | I confess I did not understand your terminologies completely. But what would be happen if one define a diffm operator of order k as a linear map $D$ such that $s\mapsto D(fs)-fD(s)$ would be of order k-1.(inductive definition). Is there a problem of GLOBAL definition of differentoal operatores in this manner? | |
| Nov 16, 2017 at 22:37 | comment | added | Pedro Lauridsen Ribeiro | Pohl's results were derived independently by Libermann (1963) (a student of Ehresmann) and Feldman (1963), but one also cannot find any explicit isomorphism there. There is a couple of references by Ehresmann (1955) and Libermann (1961) on higher-order connections I couldn't get access to (they are both in rather obscure proceedings volumes) and maybe there is something there closer to what I ask, but I really don't know. | |
| Nov 16, 2017 at 22:33 | comment | added | Pedro Lauridsen Ribeiro | @DmitriZaitsev That's true, but that's not what I am asking. I've also tracked Ehresmann's pioneer work on jets and (arbitrary-order) connections, and couldn't find any trace of the result I've stated - to wit, relating jets to iterated (first-order) covariant derivatives. The book by Jafarpour and Lewis quoted by Umberto Lupo above traces related results to a Trans. AMS paper by Pohl (1966) whose preprint actually dates back to 1963, but there (as the authors themselves state) no such formula can be found explicitly. | |
| Nov 16, 2017 at 22:16 | comment | added | Dmitri Zaitsev | The jets are usually attributed to Ehresmann | |
| Jun 20, 2017 at 14:40 | comment | added | Pedro Lauridsen Ribeiro | That is actually a very nice reference (despite lacking a bit of historical care, as you noticed and most people using/quoting the result do), thanks! | |
| Jun 8, 2017 at 10:48 | comment | added | Umberto Lupo | I had not known that the result dates back to Palais' work. I learned it from engineering.ucsb.edu/~saber.jafarpour/time_varying.pdf (see Lemma 2.1) and apparently those authors did not come across your reference either. | |
| Jul 27, 2016 at 14:10 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Added remark
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| S Jun 29, 2016 at 19:12 | history | bounty ended | CommunityBot | ||
| S Jun 29, 2016 at 19:12 | history | notice removed | CommunityBot | ||
| S Jun 21, 2016 at 17:31 | history | bounty started | Pedro Lauridsen Ribeiro | ||
| S Jun 21, 2016 at 17:31 | history | notice added | Pedro Lauridsen Ribeiro | Authoritative reference needed | |
| Jun 3, 2016 at 16:46 | history | edited | Willie Wong |
edited tags
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| Jun 3, 2016 at 15:30 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Added explanations and motivation, improved notation, a few stylistic embellishments
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| Jun 3, 2016 at 5:30 | history | asked | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |