Talk:Vertical and horizontal bundles

Pullback bundle
It seems to me that something isn't correct here:


 * The differential d&pi;:TE&rarr;&pi;*TM identifies the quotient bundle TE/VE with the pullback bundle &pi;*TM.

I think that if π:E→M, then dπ:TE→TM and not TE→π*TM.

&pi;*TM is the pullback bundle, so that d&pi; : TE &rarr; &pi;*TM is a morphism of vector bundles over E. Whereas, thinking of it as d&pi; : TE &rarr; TM, it is a mapping which covers &pi;, so that the pair (d&pi;, &pi;) is a vector bundle morphism. I have seen both in the literature, but the first way it is slightly more precise. silly rabbit ( talk ) 11:37, 13 March 2008 (UTC)


 * That's right, but you know this already. 67.198.37.16 (talk) 17:10, 22 April 2016 (UTC)

Manifold?
I am not an expert, but, are we assumming here that the top space E is a manifold or at least that the fibers over individual points are manifolds? I say this because we are referring here first of all to TE, which assumes E is a manifold, and then we refer to T_e(E_x) , where E_x is the fiber over x , i.e., we have π(e)=x , and then we consider E_x:=π^{-1}(x) , and then T_e(E_x) , so E_x must be a manifold, to have a tangent space? — Preceding unsigned comment added by 146.96.35.67 (talk) 07:58, 30 May 2013 (UTC)


 * By definition of fibered manifold, E, M are differentiable manifolds and π is a smooth map. Furthermore, one can prove that each fiber $$E_x$$ over $$x$$ is a differentiable manifold. So it makes sense to consider equivalence classes of curves $$[\gamma ]$$, with $$\gamma\colon \mathbb{R}\to E_x$$ and $$\gamma (0)=e$$, i.e., tangent vectors to the fiber $$E_x$$. See also: Talk:Connection (principal bundle) Mgvongoeden (talk) 13:08, 30 May 2013 (UTC)

Merge of Horizontal bundle
In Jan 2016, User:TakuyaMurata proposed that Horizontal bundle be merged into this article, with the note:  better to discuss the two complementary concepts at the same place; less repetition, especially.

I'm concerned about this proposal; these are related ideas, they focus on very different things. For example:
 * The vertical bundle has a gauge structure, the connection form vanishes on the horizontal bundle, and is non-zero only on the vertical bundle.


 * The solder form or tautological one-form vanishes on the vertical bundle and is non-zero only on the horizontal bundle.


 * The torsion tensor vanishes on the vertical bundle, and is used to define exactly that part that needs to be added to an arbitrary connection to turn it into a Levi-Civita connection (i.e. make a connection be torsionless.)

Doing all this ... well. Hmm. Might not be a bad idea. Changing my mind, maybe I will merge. 67.198.37.16 (talk) 17:49, 22 April 2016 (UTC)


 * I finished doing the merge. Now, this article needs to be moved to Vertical and horizontal bundles. 67.198.37.16 (talk) 20:13, 22 April 2016 (UTC)

Requested move 30 April 2016

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 

The result of the move request was: moved  KSF  T C 19:24, 18 May 2016 (UTC)

Vertical bundle → Vertical and horizontal bundles – The article covers both the vertical and horizontal bundles in a unified way, rather than each, individually 67.198.37.16 (talk) 18:13, 30 April 2016 (UTC) --Relisted. George Ho (talk) 06:12, 8 May 2016 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.