Timeline for Making the identification $\tau M\approx TM\oplus (TM\odot TM)$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 7, 2015 at 14:35 | comment | added | Michael Bächtold | Actually fibers of $\tau M$ and $T^2M$ have different dimensions, so they can't be the same. (Btw. the notation $T^2M$ is also used for the iterated tangent bundle $T(TM)$ so there is danger of confusion here.) | |
| May 7, 2015 at 13:15 | comment | added | Josh Burby | @MichaelBächtold I'm curious about this too. Lazaro Cami says that $\tau M$ is naturally a vector bundle over $M$ (which seems to make sense). Meanwhile, it looks like the arXiv paper you sent needs additional structure to give $T^2M$ a vector bundle structure. | |
| May 7, 2015 at 7:09 | comment | added | Michael Bächtold | Thanks, maybe the notions do coincide. I'm not sure yet. | |
| May 6, 2015 at 20:00 | comment | added | Josh Burby | @MichaelBächtold Sure. I first saw this use of this terminology in this PHD thesis by Joan Andreu Lazaro Cami gmcnet.webs.ull.es/files/thesis/alazaro.pdf. See p. 33. The book "Global and Stochastic Analysis with Applications to Mathematical Physics" by Yuri E. Gliklikh also uses it. On page 66 of the book, there are a number of other references. | |
| May 6, 2015 at 9:26 | comment | added | Michael Bächtold | This bundle does not seem to be the same as the ones that are usually called higher order tangent bundles like here: arxiv.org/abs/1403.3111. Could you give a reference for the notion you are using? | |
| Apr 18, 2015 at 12:57 | history | edited | Qfwfq | CC BY-SA 3.0 |
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| Apr 15, 2015 at 15:00 | vote | accept | Josh Burby | ||
| Apr 13, 2015 at 6:54 | answer | added | Matthias Ludewig | timeline score: 6 | |
| Apr 12, 2015 at 22:51 | comment | added | Josh Burby | @MatthiasLudewig Thank you. I thought it seemed very similar in spirit to the notion of a so-called Ehresmann connection. However, I didn't see how the details worked out because, in the case of the Ehresmann connection, the exact sequence is $V_eE\rightarrow T_eE\rightarrow T_{\pi(e)}B$, where $\pi:E\rightarrow B$ is some fiber bundle and $VE\subset TE$ is the vertical subbundle. I couldn't see how $\tau_m M$ plays the role of $T_eE$. | |
| Apr 12, 2015 at 20:56 | comment | added | Robert Bryant | More precisely, specifying such a splitting is equivalent to specifying a torsion-free (aka symmetric) connection. | |
| Apr 12, 2015 at 20:41 | comment | added | Matthias Ludewig | This is usually called a connection. This may be somewhat confusing, as there are other objects called connection (connections as certain first order differential operators, connections as certain lie-algebra-valued one-forms on principal bundles), but it all makes sense as all these things are equivalent to each other. | |
| Apr 12, 2015 at 20:37 | history | asked | Josh Burby | CC BY-SA 3.0 |