Talk:Connection form

Change of frame[edit]

What is meant by ″applying the exterior connection to both sides″ in the paragraph below? How is $\omega (\mathbf {e} ')=g^{-1}dg+g^{-1}\omega (\mathbf {e} )g$ ?

In order to extend ω to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of E is chosen. Write ω_α^β = ω_α^β(e) to indicate the dependence on the choice of e.

Suppose that e′ is a different choice of local basis. Then there is an invertible k × k matrix of functions g such that

{\mathbf {e} }'={\mathbf {e} }\,g,\quad {\text{i.e., }}\,e'_{\alpha }=\sum _{\beta }e_{\beta }g_{\alpha }^{\beta }.

Applying the exterior connection to both sides gives the transformation law for ω:

\omega (\mathbf {e} \,g)=g^{-1}dg+g^{-1}\omega (\mathbf {e} )g.

Wisapi (talk) 18:36, 29 February 2024 (UTC)[reply]

other homie[edit]

The exterior covariant derivative is a very useful notion which makes possible to simplify formulas in using connection.

Given a tensor-valued differential k-form  $\phi$  its exterior covariant derivative defined by
 $D\phi (X_{0},X_{1},...,X_{k})=d\phi (h(X_{0}),h(X_{1}),...,h(X_{k}))$ 
where h denotes the projection to the horizontal subspace,  $H_{x}$  with kernel  $V_{x}$ .

I don't understand this. Is X the position and the X_i's the coordinates? But that can't be because it should be h(X₁,...,X_k) then. And how is X related to x? Is the "tensor" φ takes values in a rep of G? Also, if two points project to the same point in the base space, are the values of φ at these two points related or independent? Phys 19:29, 14 Aug 2004 (UTC)

Is φ defined over E or B? Phys 19:44, 14 Aug 2004 (UTC)

Hope its better now Tosha 21:07, 15 Aug 2004 (UTC)

But one thing I still don't understand is in your examples with vector bundles instead of principal bundles, φ is defined over B and not E. Phys 06:10, 17 Aug 2004 (UTC)

See the first par in vector bundles. Tosha 23:56, 21 Aug 2004 (UTC)

Well done[edit]

Nice article, but it lacks any reference to the second Bianchi identity

D\Omega =0

which is true for the exterior covariant derivative for any connection in a principal bundle. Also, as the article defines it, torsion only applies to affine connections. In which case, you should also find some place for the identity

D\Theta =[\Omega ,\theta ].

Silly rabbit 04:56, 8 November 2005 (UTC)[reply]

I just added a very quickie brief mention of the Bianchi identities. I guess this could be unpacked a bit.67.198.37.16 (talk) 18:12, 23 October 2016 (UTC)[reply]

Connections and Jet Bundles[edit]

A connection in a principal bundle is not the same thing as a smooth section on the bundle JE -> E. One, the latter is more general and (in fact) is not specific to principal bundles. Two, the sections corresponding to the connection in a principal bundle is equivariant under the action of the group. This latter condition is what's missing.

The more general concept of a connection as a section over the jet bundle applies generically -- not just to principal connections, or connections inherited by bundles associated to principal bundles (both called principal connections), but to bundles in general. In the former two cases, the more general connection need not coincide with an existing principal connection. The two will differ by what is sometimes called a "soldiering" form.

The mention of jet bundles and connections, therefore, should be brought out in a separate section indicating also that it also applies to non-linear bundles, not just to vector and principal bundles. -- Mark, 11 November 2006

Factors of two[edit]

There are three different ways to write the quadratic term in the structure equation:

d\omega +[\omega ,\omega ]=d\omega +\omega \wedge \omega =d\omega +{\tfrac {1}{2}}[\omega \wedge \omega ].

In the first, the skew symmetry of the Lie bracket implies [ω,ω] is a 2-form. In the second, we have to assume ω takes values in a representation (e.g. it is a matrix, or is in the universal enveloping algebra) so that the skew symmetry of the wedge product implies that this term is a commutator, hence a Lie bracket. In the final term, the wedge product is contracted by the Lie bracket: here everything is explicitly in the right place, but we have to divide by two. Further explanation available on request! Geometry guy 21:45, 14 February 2007 (UTC)[reply]

How are you defining [ω,ω]? I've always thought this was synonymous with [ω∧ω]. At least it is in all the books I've checked. At any rate, I'd vote in favor of using the notation [ω∧ω] everywhere as it is the most explicit. -- Fropuff 20:59, 16 February 2007 (UTC)[reply]

There is some ambiguity here about how the 1-forms are multiplied, but my convention is that, unless indicated otherwise, use tensor product, i.e., [ω,ω]_X,Y = [ω(X),ω(Y)]: this also agrees with what would happen in index notation. Anyway, I agree that ambiguity-free notation is much more preferable. Geometry guy 00:54, 18 February 2007 (UTC)[reply]

I see. So in that notation, given two Lie algebra-valued forms ω and η, the bracket [ω,η] would not be a alternating form (in general). I've not seen the bracket used that way but I guess it make sense. (I note that curvature form also uses [ω,η] to mean [ω∧η].) At any rate, I've briefly explained the [ω∧η] notation over at Lie algebra-valued form so perhaps we can start using that notation and referencing that article as appropriate. -- Fropuff 15:55, 18 February 2007 (UTC)[reply]

Many moons ago, I tried (unsuccessfully, in my opinion) to bring some order to the wedge and bracket conventions used on Wikipedia. At some point, I must have had it all sorted out (in my own mind, at least). The "standard" definition in differential geometry (if there is such a thing, at all; I've seen both conventions) is to use the skew-part of a tensor. The Wikipedia convention is the rather more exotic (k,m)-shuffles version, selected in part because it works over fields of finite characteristic. I suspect that many of my early efforts to correct all of the factors of two have been trampled (not that I blame the editors for doing so).

The upshot is that, pace Geometry Guy, there is unambiguously a factor of 1/2 in the structural equations:

d\omega +{\tfrac {1}{2}}[\omega ,\omega ]=d\omega +\omega \wedge \omega =d\omega +{\tfrac {1}{2}}[\omega \wedge \omega ].

I think there are good reasons for this, apart from being just a tool for simplifying formulas. The bracket operation is, in effect, coupled to the wedge product of forms. (On a Lie group, this agrees with the usual graded "super"-commutator of graded derivations on the algebra of differential forms.) Furthermore, the bracket can be defined for forms of higher degree, in which case the compatibility conditions become important (see Frolicher-Nijenhuis bracket for hints in this direction.) And, let's not leave out the most obvious: it's a matrix commutator if you want it to be!

Anyway, I can see the source of confusion, and it's quite understandable. Look at the way Kobayashi-Nomizu define the bracket of forms. This is precisely the way G.G. would have it. But, they're also using the skew-part definition of the wedge product, so they have a factor of 1/2 in their formulas too. I think it's important to stick to one convention. It might be worth writing a specification of these conventions (as well as a detailed explanation — properties, etc. — of why this convention was settled upon.) My 2¢, Silly rabbit 00:47, 4 May 2007 (UTC)[reply]

Overlap with Ehresmann connection[edit]

This page overlaps a lot with Ehresmann connection. I've added a link, but I think an opportunity is being missed here. The term connection form is often used for the 1-form defining a connection (e.g. on a vector bundle) relative to a frame. This article could begin with such a naive point of view (linking e.g. gauge covariant derivative) and then explain that the frame-dependent notion of a connection form becomes well defined when it is lifted to the frame bundle. This actually motivates the idea of an (Ehresmann) connection on a principal bundle (the horizontal space is the kernel of the connection form), hence on general fiber bundles, in particular vector bundles, which brings us back to the original motivation. Geometry guy 21:52, 14 February 2007 (UTC)[reply]

I am somewhat unsatisfied with the organization of the connection articles on Wikipedia. I've long felt that we should have a page devoted to connections on principal bundles and one to connections on vector bundles. Probably at connection (principal bundle) and connection (vector bundle) respectively (both currently redirects). The principal connection material is currently spread between this page and Ehresmann connection while the corresponding material for vector bundles is distributed between here and covariant derivative. I've started a draft organizing the vector bundle material at User:Fropuff/Draft 12 which I plan to move into the main namespace sometime soon (comments welcome). I believe the principal connection material deserves a similar treatment. As you rightly point out, the connection form idea intertwines many of these ideas and this page could be rewritten accordingly. -- Fropuff 07:22, 15 February 2007 (UTC)[reply]

I'm also rather unsatisfied with the current organization of the connection articles. There are many different ways to say the same thing, but at present the relationships between the different approaches are not properly spelt out and there is much duplication. In particular, a connection form in the sense of "vertical valued 1-form on the total space of a fibre bundle" is just a way of saying "Ehresmann connection". (As an aside, the prefix "Ehresmann" is only used these days to emphasise that the connection is not a principal or vector bundle connection!)

I think the new pages you propose are a step in the right direction and the draft looks good. Concerning terminology, I believe that very few differential geometers say "Koszul connection" these days, prefering "connection on a vector bundle" or "covariant derivative". Although I see some sense in restricting the use of the term "covariant derivative" to tensor bundles (as you seem to be suggesting), this does not agree with common usage. Indeed, many (including myself) think of a connection as the object (connection form or horizontal subspaces) defining infinitesimal parallel transport, and the covariant derivative is the corresponding differential operator: in particular, the term exterior covariant derivative is widespread for the exterior derivative coupled to a connection on a vector bundle. Geometry guy 10:59, 15 February 2007 (UTC)[reply]

Looking at your draft a bit more carefully, I see you have already covered most of these points! So, I guess it is just the first sentence that is a bit misleading. Geometry guy 11:02, 15 February 2007 (UTC)[reply]

Exterior connection[edit]

What is an exterior connection? The term is not actually defined in the article, and Google turns up no other references using this terminology. The article would be clearer without such nonstandard terminology (which I feel is probably unnecessary anyway). 198.84.187.170 (talk) 21:49, 21 January 2014 (UTC)[reply]

Well, it is clearly defined as being the same thing as the Exterior covariant derivative; however, I don't really know if this is a wikipedia neologism, or is in use in various references. I just loooked in my three favorite references, and it does not appear in the index. (MTW, Marsden, Jost) 67.198.37.16 (talk) 18:21, 23 October 2016 (UTC)[reply]

Confused by definition of "local frame"[edit]

A local frame for E is an ordered basis of local sections of E.

So what kind of object is it? Is a local frame defined for E, or for a given neighborhood, or for a given point? The collection of local sections isn't a vector space, so it can't have a basis. --146.96.28.240 (talk) 23:29, 6 May 2015 (UTC)[reply]

The local frame is given w.r.t. the atlas (topology), and in that case, each fiber is a vector space that is just Euclidean space, and has a well-defined, ordered basis. aka local trivialization. I just added to the article. 67.198.37.16 (talk) 18:28, 23 October 2016 (UTC)[reply]

Notation between solder $\theta$ and relation to Christofel symbol[edit]

The $\theta$ used in the solder form discussion may not be the same $\theta$ used in the immediately following example concerning the Christofel connection. Since a good thing to do is make very clear how the frame language relates to more familiar/first-learned Cristofel symbol. Could someone please clarify 2602:306:CC4B:2980:4D3D:BAC4:E99F:19EB (talk) 16:10, 1 July 2016 (UTC)[reply]

Well, but it is the same thing, although it requires a slight abuse of notation. A bit later in the article, it says so explicitly; I quote: "the solder form is θ = Σ_i e_i ⊗ θⁱ, where again θⁱ is the dual basis." The solder form is the identity endomorphism from the vector space to itself. 67.198.37.16 (talk) 19:14, 23 October 2016 (UTC)[reply]

Which property?[edit]

Under "Exterior connections", is mentioned that D can be extended to the full exterior algebra of differential forms if "it satisfies this property". However, such a property is nowhere to be found. In the main article, it says that to extend D it is sufficient to impose the Leibniz rule for graded derivations and commuting with the insertion operator. Could this second property be the lost one? Mattecapu (talk) 17:23, 4 February 2019 (UTC)[reply]