Talk:Ehresmann connection

Definition

Why $d\pi :TE\rightarrow \pi ^{*}TM$ and not simply $d\pi :TE\rightarrow TM$? Noix07 (talk) 17:11, 9 November 2014 (UTC)

In "definition via the connection form" doesn't v²=v already implies that it is identity on the image? and 0 on its kernel? Noix07 (talk) 17:11, 9 November 2014 (UTC)

Yes, I agree v²=v already implies that it is identity on the image but not that this image is V. Nevertheless, the assertion "v is the identity on V=Image (v)" will imply v²=v. So, when it is said that the two conditions imply that v defines a connection I would add that the second condition already implies the first one. 150.214.18.71 (talk) 14:12, 4 February 2016 (UTC)

Vector bundles and covariant derivatives

I don't understand this section. I'm afraid that there is some mismatch in the usage of e. If e ∈ E, then it makes no sense to multiply it or translate by it. In other hand, if e is an element of the fiber, then the multipication and addition is defined on the fiber only and not on E. More precisely, these operations on E_x (as a subset of E) depend on the local trivialization of the bundle. —Preceding unsigned comment added by 86.101.197.154 (talk) 07:03, 20 February 2008 (UTC)

The fibers in a vector bundle can be given the structure of a vector space in natural fashion. This does not depend on the local trivializations as you suggest. This follows from the fact that the transition functions are linear isomorphisms (or that the structure group is GL(n)). Compare definitions 1 and 2 in the vector bundle article. It's a good exercise to show that these are equivalent. -- Fropuff (talk) 00:22, 21 February 2008 (UTC)

Ok, Take two local trivializations with transfer function L. Suppose that in the first trivialization e is mapped to (x,f) and in the second trivialization to (x,L(f)). Multiplying e by λ in the trivializations yield (x,λ f) and (x, λ L(f)) = (x, L(λ f)). These are really the images of the same element in the given two trivializations.

Take another point p in this fiber. The image of it in the trivializations are (x,g) and (x,L(g)) respectively. Shifing p by e in the first trivialization yields (x,g+f) while in the second one (x,L(g)+L(f)) = (x,L(g+f)). The result is the same again. You are right, thank you. —Preceding unsigned comment added by 86.101.197.154 (talk) 06:25, 21 February 2008 (UTC)

I created a new article Secondary vector bundle structure which contains a section on Ehresmann connections on vector bundles, and something about the linearity of a connection. Since this article is more elaborated, I will not tamper with it without warning, but I think some kind of interlinking would be in place. Lapasotka (talk) 14:10, 19 February 2010 (UTC)

Could somebody tell me, what is the differene between $H\boxtimes H$ and $H\times _{M}H=\{(v,w)\in H\times H:\pi (v)=\pi (w)\}$? --89.135.29.163 (talk) 07:28, 26 October 2010 (UTC)

I think that the author of this paragraph means

$H\boxtimes H=H\times _{M}H:=\{(x,v,w,X,Y):X\in H_{(x,v)},\ Y\in H_{(x,w)},\ (x,v)\in E_{x},\ (x,w)\in E_{x},\ x\in M\}.$ Lapasotka (talk) 22:51, 26 October 2010 (UTC)

OK, thanks. But isn't more simple to say that we take the bundle $\pi :H\rightarrow M:X\mapsto x$ where $X\in H_{(x,v)}$ and $(x,v)\in E_{x}$ and then we can omit x,v and w from your set and write simply $\{(X,Y):X\in H,Y\in H,\pi (X)=\pi (Y)\}\,$. And this is exactly my interpretation except we wrote X instead of v and Y instead of w. --89.135.29.163 (talk) 06:33, 27 October 2010 (UTC)

Note that your interpretation of $H\times _{M}H$ is a little bit off. Check also Secondary vector bundle structure for an alternative explanation of this topic. Lapasotka (talk) 22:51, 26 October 2010 (UTC)

Thanks, this would be probably helpful, but maybe unnecessary complication for understanding this definition. I think that it would have been the best to say simply that the connection is linear by defintion if parallel transport preserves linear combination, isn't it? --89.135.29.163 (talk) 06:33, 27 October 2010 (UTC)

I am a bit worried because this condition does not appear in classical Walter A. Poor's book "Differential Geometric structures" McGraw Hill (1981), see the definition 2.26 in that reference. The key to obtain linearity in Poor's is that any smooth homogeneous mapping on a vector space must be linear (essentially, it could be identified with its differential at 0). By the way, it is not clear for me the definition of completeness in the article, as it would not match with the usual notion of completeness for a Riemannian manifold (which is defined in terms of the extendability of its geodesics). A different notion of completeness relies on evolutes, as explained by Lumiste for Cartan connections https://www.encyclopediaofmath.org/index.php/Connections_on_a_manifold (however, this seems to be different to the notion of completeness in the article). 150.214.18.71 (talk) 19:39, 5 February 2016 (UTC)

What are the restrictions on the bundle?

Currently the definition of an Ehresmann connection is given with respect to an arbitrary fiber bundle, which is a purely topological space. This doesn't seem fair, given that the definition makes reference to tangent vectors of the fiber, base, and total space. In other words, don't these spaces need to be differentiable manifolds at the very least? Trevorgoodchild (talk) 02:45, 8 May 2010 (UTC)

I added the requirement "smooth". I don't believe it makes sense to chase the most general definition in this article. Lapasotka (talk) 12:03, 8 May 2010 (UTC)

Definition via horizontal subspaces

In "Definition via horizontal subspaces", there is the following remark: In more sophisticated terms, such an assignment of horizontal spaces satisfying these properties corresponds precisely to a smooth section of the jet bundle J¹E → E.

Should this really be any smooth sections of the Jet bundle, or rather those corresponding to fiberwise injective linear Maps (under the indentification with an appropriate subbundle of Hom(TM,TE)? — Preceding unsigned comment added by 137.250.27.7 (talk) 14:30, 22 February 2019 (UTC)