Talk:Vertical and horizontal bundles

Pullback bundle

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

The differential dπ:TE→π*TM identifies the quotient bundle TE/VE with the pullback bundle π*TM.

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

π^*TM is the pullback bundle, so that dπ : TE → π^*TM is a morphism of vector bundles over E. Whereas, thinking of it as dπ : TE → TM, it is a mapping which covers π, so that the pair (dπ, π) 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 (non-admin closure) KSFT^C 19:24, 18 May 2016 (UTC)

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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)

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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.