Talk:Connection form

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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 φ its exterior covariant derivative defined by
Dφ(X0,X1,...,Xk) = dφ(h(X0),h(X1),...,h(Xk))
where h denotes the projection to the horizontal subspace, Hx with kernel Vx.

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)

Contents 1 Well done 2 Connections and Jet Bundles 3 Factors of two 4 Overlap with Ehresmann connection

[edit] Well done

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

DΩ = 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

DΘ = [Ω,θ].

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

[edit] Connections and Jet Bundles

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

[edit] Factors of two

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

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)

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)

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)

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)

[edit] Overlap with Ehresmann connection

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)

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)

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)

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)