Talk:Double tangent bundle

There exists a characterization/definition

Maybe someone could replace "The Lie Bracket of two vector fields on a manifold also has a formulation in terms of the double tangent bundle." with an actual formulation, unless it is extremely obscure or long. Lapasotka (talk) 13:01, 22 August 2009 (UTC)

I can write that up. It's pretty much a "follow your nose" issue. Rybu (talk) 09:47, 19 November 2009 (UTC)

Connections in this article

There is a red link into Nonlinear covariant derivative in Finsler manifold. Someone seems to have ideas for treating connections in this article, so I will not touch the topic for a while. Nonlinear connections on the tangent bundle have to be treated somewhere, maybe on this page. They are desperately needed in Finsler manifolds and spray structures. —Preceding unsigned comment added by Lapasotka (talk • contribs) 17:18, 22 August 2009 (UTC)

the link is in the double tangent bundle article, here: Ehresmann_connection Rybu (talk) 09:47, 19 November 2009 (UTC)

Relation to jet bundles?

I would love it if someone knowledgeable would fill in the details about how T^2(M) is related to jet bundles (which I do not understand very well). Kier07 (talk) 06:30, 19 November 2009 (UTC)

I deleted the comment on jet bundles, because it was probably more confusing than enlightening. For example, the second order jet bundle on n-dimensional manifold M has rank 1+n+n(n+1)/2 and base space M, whereas TTM has rank 2n and base space TM. The only simple relationship between double tangent bundles and second order jet bundles I can think of is the fact that for any $f,g:M\to N$ the second order jets $J^{2}f,J^{2}g\in J^{2}(M,N)$ coincide if and only if the iterated tangent mappings $f_{**},g_{**}:TTM\to TTN$ coincide. In my opinion this relationship is not simple enough to be stated in the introduction without any explanation. A full section which discusses the iterated tangent mappings could contain this remark. Lapasotka (talk) 05:12, 18 February 2010 (UTC)

Fixed relation involving canonical endomorphism J

The "identity" $J[X,Y]=J[JX,Y]+J[X,JY]$ appeared in the article, which is false. I have replaced it with the correct identity $[JX,JY]=J[JX,Y]+J[X,JY]$, which since the Nijenhuis tensor $N_{J}$ of $J$ is $[JX,JY]-J[JX,Y]-J[X,JY]$, is equivalent to the claim that $N_{J}=0$. The article Tangent bundle geometry Lagrangian dynamics (cite M Crampin 1983 J. Phys. A: Math. Gen. 16 3755) is a reference.

The original "identity" must be false, since it implies for any $X$ and $Y$ that (since $J^{2}=0$)

$J[X,Y]=J[JX,Y]+J[X,JY]=J[J^{2}X,Y]+J[JX,JY]+J[JX,JY]+J[X,J^{2}Y]=2J[JX,JY]=2J[J^{2}X,JY]+2J[JX,J^{2}Y]=0$

which of course means $J=0$. —Kphoek (talk • contribs) 08:34, 27 February 2024 (UTC)