Talk:Connection form

message: 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 ωαβ = ωαβ(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^\beta_\alpha.
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)
thread_title:  Change of frame 

message: 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 Xi's the coordinates? But that can't be because it should be h(X1,...,Xk) 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)
replies: text: Is φ defined over E or B? Phys 19:44, 14 Aug 2004 (UTC)

text: 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)
thread_title:  other homie 

message: 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)
replies: text: 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)
thread_title:  Well done 

message: 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+\tfrac12[\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)
replies: text: 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)
replies: text: 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)
replies: text: 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)

text: d\omega+\tfrac12[\omega,\omega]=d\omega+\omega\wedge\omega=d\omega+\tfrac12[\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)
thread_title:  Factors of two 

message: 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)
replies: text: 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)
thread_title:  Exterior connection 

message: :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)
replies: text: 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)
thread_title:  Confused by definition of "local frame" 

message: 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)
replies: text: 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 = Σi ei ⊗ 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)
thread_title:  Notation between solder <math>\theta</math> and relation to Christofel symbol 

message: 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)
thread_title:  Which property?