Timeline for The bundle of symmetric affine connections as quotient of the second-order frame bundle
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 9, 2018 at 10:17 | vote | accept | Giovanni Moreno | ||
| Apr 9, 2018 at 10:17 | comment | added | Giovanni Moreno | P.S.: now that the keyphrase "second-order frame" has appeared, I found the post mathoverflow.net/questions/279239/… which I did not notice before because I was looking for "space of affine connections". Indeed, such a space is described (the way I wanted it to be described) in Section 17.7 of the book by Kolar, Michor & Slovak. Thank you for the enlightening discussion. | |
| Apr 9, 2018 at 9:59 | comment | added | Giovanni Moreno | Concerning the comment, your $F^*(M)$ is precisely what I've denoted by $\hat{J}^1_0(M,\mathbb{R}^n_0)$ and called erroneously "frame bundle" (instead of coframe). My formula ($*$) is precisely $J^1(F^*(M))/\mathrm{GL}_n(\mathbb{R})$ of yours. And what I denoted by $\hat{J}^2_0(M,\mathbb{R}^n_0)$ now corresponds to your $J^1_0(F^*(M))$. That is, "2-nd order coframings'' (provided such a terminology exists) can be seen as 1-jets of closed coframings. This indeed makes things more transparent, for all boils down to the fact that $J^2$ is the subspace of $J^1(J^1)$ of ''holonomic'' elements. | |
| Apr 7, 2018 at 21:48 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Added a remark about another way to think about torsion-free affine connections.
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| Apr 7, 2018 at 21:23 | comment | added | Robert Bryant | If it were me having to use this fact in a paper or a lecture, I would simply put in a remark describing the identification (as I did) and say (as I did) that the reader can readily check, via local coordinates, that it has the desired properties. I might add a footnote along the lines of "I am not sure to whom this construction should be attributed, and I would be interested to know a reference to its first explicit use." While the construction is of historical interest, I think it's not a significant enough result to warrant a lot of concern about attribution. | |
| Apr 7, 2018 at 19:34 | comment | added | Giovanni Moreno | Excellent. Now I only need a team of french-speaking post-docs working around the clock trying to find a needle in a haystack. | |
| Apr 7, 2018 at 17:43 | comment | added | Robert Bryant | In case you want to look through Ehresmann's collected works to try to find a reference there, they are available online for free at ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/C.E_Works.htm | |
| Apr 7, 2018 at 17:16 | comment | added | Robert Bryant | Have you tried looking at Ehresmann's papers? I certainly expect that either it is done somewhere there or a reference is given. | |
| Apr 7, 2018 at 15:36 | comment | added | Giovanni Moreno | Of course you're right. My reasoning (which I didn't post) was the following: define $\gamma(\Delta)_m$ directly by letting $[x]^2_{m,0}\cdot\mathrm{GL}_n(\mathbb{R})$, where $x:U_m\to\mathbb{R}^n$ is an inertial reference frame at $m$, that is a coordinate system where the Christoffel symbols of $\Gamma$ vanishes at $m$. I think this "my" definition is equivalent to yours. And now I'm sure things work out the way they should. Still, it's frustrating not being able to find such a trifle in the literature. "Folklore" cannot be cited! I'll leave the post open, maybe somebody has a reference | |
| Apr 7, 2018 at 10:58 | history | answered | Robert Bryant | CC BY-SA 3.0 |