Talk:Vertical and horizontal bundles

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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)[reply]

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

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)[reply]

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}

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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.