Talk:Connection (vector bundle)

message: As Connection (vector bundle) seems to cover curvature, too, I'll ask here.
Is there a formula like:
Assume there is a smooth family of smooth curves \gamma _t(s) with \gamma _t(0)=p and \gamma _t(1)=q for some p and some q, points in a manifold M. Let E be a vector bundle over M with a connection. Let P_{t,s_1,s_2}:E_{\gamma_t(s_1)}\to E_{\gamma_t(s_2)} be the parallel transport along \gamma _t for s_1 to s_2. Then one has for some V\in E_p
\frac{d}{dt}P_{t,0,1} |_{t=0} V = \int_{0}^1 P_{t,s,1} |_{t=0} R(\frac{d}{ds'}\gamma _t(s')|_{s'=s},\frac{d}{dt'}\gamma _{t'}(s)|_{t'=T})P_{t,1,t} |_{t=0}Vds
I can't find it, but it seems interesting to me, in order to understand the connection between curvature and parallel transport along different curves. One could then integrate over t or leave the V or take p=q.
Am I missing out on something? Is this in every text book and I'm just overlooking it. Or is it not in every textbook, because it's maybe both wrong and irrelevant?
Thanks, JanCK 17:28, 1 March 2007 (UTC)
replies: text: You may be able to get something (in case you don't already know) from the physics approach to parallel transport via Wilson lines and path-ordered exponentials. Geometry guy 21:51, 20 March 2007 (UTC)

text: You may want to have a look at [this article] titled "How the Curvature Generates the Holonomy of a Connection in an Arbitrary Fibre Bundle" by Helmut Reckziegel and Eva Wilhelmus. --131.211.23.114 (talk) 12:23, 21 June 2011 (UTC)
thread_title:  curvature - change in parallel transport 

message: I am rather confident the formula
:\nabla\sigma = (\mathrm d\sigma^\alpha + {\omega^\alpha}_\beta)e_{\alpha}
should read
:\nabla\sigma = (\mathrm d\sigma^\alpha + {\omega^\alpha}_\beta \sigma^\beta)e_{\alpha}
As I am not an expert I preferred not to immediately replace this.
The same problem occurs at the end of this paragraph.
edit: edited 84.192.174.151 12:35, 11 August 2007 (UTC)
thread_title:  Local expression 

message: It seems that there is a considerable overlap in the terminology related to connections on smooth manifolds and smooth fibre bundles. It would require a concerted effort to bring all these definitions in order and properly interlinked between the different pages. I don't want to start an edit war by single handedly forcing a naming convention for these objects, but I propose the following:
* Connection on a smooth fibre bundle (E,p,M) should refer to an Ehresmann connection TE=HE⊕VE.
* Connection on a smooth manifold M should refer to an Ehresmann connection on (TM/0,πTM/0,M).
* The corresponding nonlinear covariant derivative on a smooth manifold M into the direction of X∈TM/0 should refer to the differentiation DX in the C∞(M)-module of smooth vector fields on M determined by
:
D:(TM\setminus 0)\times \Gamma(TM) \to TM \quad ; \quad D_XY :=  (\kappa\circ j)(Y_*X),
:where κ:T(TM/0)→TM is the connector map and j:TTM→TTM is the canonical flip.
* Linear connection on a smooth vector bundle (E,p,M) should refer to an Ehresmann connection satisfying the extra property that the vertical and horizontal projections vpr:TE→VE and hpr:TE→VE are linear with respect to the secondary vector bundle structure (TE,p*,TM).
* The corresponding linear covariant derivative on a smooth vector bundle (E,p,M) into the direction of X∈TM should refer to the differentiation ∇X in the C∞(M)-module of smooth sections of (E,p,M) determined by
:
\nabla:TM\times\Gamma(E)\to\Gamma(E) \quad ; \quad \nabla_Xv := \kappa(v_*X),
:where κ:TE→E is the connector map.
This terminology is compatible with principal connections and it does not rule out non-linear covariant derivatives on smooth manifolds, which are indispensable in the study of spray structures and Finsler-spaces. Note also that the term nonlinear covariant derivative does not make sense on general vector bundles, since there an Ehresmann connection induces a differentiation for the sections if and only if it is a linear Ehresmann connection. Lapasotka (talk) 16:07, 10 March 2010 (UTC)
replies: text: Oppose.  Unfortunately, this overlap is the norm not just on Wikipedia, but throughout mathematics and physics.  Typically, when one says "connection in a vector bundle", one means what you would call a "linear connection".  However, the term "linear connection" also often means what others would call an "affine connection"; that is, a connection in the tangent bundle.  Now, obviously, one can define a connection in a fiber bundle in a fairly general way (an Ehresmann connection), and if that fiber bundle happens to be a vector bundle, then one obtains a general connection (what the Finsler people would call a nonlinear connection) in the bundle.  But this article is about the former, more restricted notion of "connection in a vector bundle", rather than the latter more general one.  I think that this focus is justified, since most mathematicians hearing the term "connection in a vector bundle" will automatically think of a linear connection.  (Going even further, many mathematicians hearing the term "connection" will probably think of linear connections only.)  In fact, even in the Finsler literature there is enough of an ambiguity in the use of the term "connection" to warrant the introduction of the standard term "nonlinear connection" to refer to the general case.  For these reason, I am opposed to any mandate that the term "connection" must necessarily refer to the latter, more general case.  However, I do think that an article about nonlinear connections is certainly warranted. Sławomir Biały (talk) 14:06, 28 March 2010 (UTC)
replies: text: I agree that in this article as well as in the majority of connection-related articles the focus should be in linear connections. However I think it is misleading to have statements such as "If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear.″ in the introduction. My main point was to clarify the distinction between connections and covariant differentiation, which might cause confusion to people who probably know Riemannian geometry and have stumbled into some more general definitions of connections. This group is probably the one which benefits the most from Wikipedia's connection-related pages. Keeping this in mind, in my opinion it is better to encourage a systematic style than to portray the typical way of communicating ideas between people more familiar with the subject matter. The latter have the ability to "read between the lines" and extrapolate all relevant implicit assumptions, but for newcomers this style of exposition can be quite frustrating. You are also correct that the ambiguity in this issue is not a fault in Wikipedia but an unfortunate path the history has taken in differential geometric literature. Still I think the optimal solution would be to have one page which explains how the term "connection" has been used in different contexts, and the rest of the connection-related pages should be written with some coherent naming convention. Lapasotka (talk) 01:30, 29 March 2010 (UTC)
replies: text: There is the connection (mathematics) that does cover various different kinds of connections, albeit emphasizing the linear case.  I'm not sure I care much for the long example there, but presumably it serves some motivational purpose.  The naming conventions seem as coherent as they can be reasonably expected to be in a general purpose reference.  "Connection in a vector bundle" refers to linear connection, "principal connection" refers to an equivariant connection in a principal bundle, "Cartan connection" refers to an Ehresmann connection with soldering.  The bare word "connection" means different things depending on the context, and I am chiefly opposed to an attempt to mandate that the term must refer to the most general possible thing.  Each article should itself be responsible for building the context in which this word is meaningful.  Thus a connection for the connection (principal bundle) can and should mean something very different from a connection for the connection (vector bundle) article.  Also, the term "linear covariant derivative" seems to me something of a pleonasm: by far the vast majority of sources take linearity as part of the definition of covariant derivative.  While there may be nonlinear versions of the covariant derivative in very special application areas, I believe these are best distinguished by prefacing them with "nonlinear", rather than insisting that every other covariant derivative must necessarily be called a "linear covariant derivative".  Sławomir Biały (talk) 11:53, 29 March 2010 (UTC)

text: See how you use the term "Principal connection" instead of "Connection on a principal bundle". It would be completely analogous and more economical to use "Linear connection" instead of "Connection on a vector bundle", and this naming would not rule out the connections that do not respect these extra structures. My main argument was that a covariant derivative should refer to a differentiation (linear or nonlinear w.r.t. direction on the base manifold) on the module of vector fields, as even the term itself suggests, whereas connection should refer to the horizontal-vertical decomposition, which makes sense for general fibre bundles. The reason why these two concepts are usually mixed up is of course that linear Ehresmann connections and linear covariant derivatives can be identified, but this is not true for nonlinear ones, except on the tangent bundle, where one can apply the canonical flip. I think it would be worthwhile to point out this source of confusion very clearly, and it would be best done by sticking with some standard (i.e. not novel and non-standard) terminology that automatically takes care of this distinction. One plausible solution would be to reserve "covariant derivative" without prefixes to linear covariant derivatives and "connection" without prefixes for nonlinear connections. This would also emphasize the important role of linearity in the identification of these two different concepts. I am not looking for a strict mandate on how things should be done in Wikipedia, but rather like to hear the opinions of other people who would potentially contribute to the related pages. Since you have been far more active in this community, you might know more people who would like to contribute to this discussion. Lapasotka (talk) 14:29, 29 March 2010 (UTC)
replies: text: I have approached User:Geometry guy, who might have some meaningful input to offer.  User:Fropuff seems to be no longer very active, and User:Silly rabbit is retired.  Anyway, I think both of our points have been made.  I would also be curious to see what others think.  Sławomir Biały (talk) 16:56, 29 March 2010 (UTC)
thread_title:  Organization of connection-related material 

message: Substituted at 01:55, 5 May 2016 (UTC)
thread_title: Assessment comment

message: There is some annoying inconsistency in WP regarding the use of nabla \nabla vs. capital D. So, this article uses nabla, consistently, everywhere. However, the article connection form uses D for the general case, and switches to nabla for Levi-Civita. (This is consistent with what Jost does in his  Riemannian geometry book) The article Yang-Mills theory uses capital D. My experience is that nabla is an older notation, and tends to be used only for Levi-Civita, and not for connections in general, while D is used for the general case. Is it reasonable to attempt some sort of standardization?
Speaking of which: there are some more conflicts of notation-- this article, as well as connection form both use \omega as the connection form, even though this conflicts with \omega as a generic alternating form on the vector bundle (see Vector-valued differential form, which this article invokes).  By contrast, the Yang-mills article (and Jost) use A for the connection form. Also, Yang-Mills and Jost use F for the curvature, instead of \Omega, which then frees up \Omega to indicate the space of differential forms.  My personal preference is for A and F over \omega and \Omega, since, again, I get the impression that \omega and \Omega is an older notation reserved for Riemannian manifolds, rather than for vector bundles.  I'm interested in trying to have a bit more standardization here, and doing it in a way that is consistent with modern usage.  67.198.37.16 (talk) 21:40, 23 October 2016 (UTC)
thread_title:  Nabla vs. D 

message: In the diagram connecting parallel transport and covariant derivative, the lablel "s(x)" should be replaced by the label "s(0)=x", I believe.  — Preceding unsigned comment added by 77.198.158.75 (talk) 13:40, 12 January 2021 (UTC)
replies: text: The label of s(x) is correct. s(0) doesn't make sense as s is a map from the base manifold M so its argument has to be a point (such as x, \gamma(t_1), etc). I suppose you might mean that the label for the point x should be replaced by x = \gamma(0), but this is explained in the caption and I felt it would have cluttered up the already reasonably complicated diagram.Tazerenix (talk) 00:40, 13 January 2021 (UTC)
thread_title:  Errata 

message: The section titled Local expression may be absolutely correct, but it could with just a little more work be much Much MUCH clearer.
1. For one thing: How about NOT using the Einstein convention and use summation sigmas instead, with the range of subscripts written out in full rather than "understood"?
That would be a start.
2. Also, try to eventually express things without any tensor product notation.
3. For another improvement: Express the connection in local coordinates as a function in standard notation: Show its symbol, followed by a colon, then its domain, followed by a right arrow, then its codomain. Then display this function as operating on a typical element of its domain, and follow that with an equals sign, and on the right of the equals sign display how this function is evaluated.
4. Do not use obscure terms like "saturating the subscript", because very few readers will understand them. 2601:200:C000:1A0:4C3C:C3A:335A:DC6 (talk) 19:19, 9 February 2021 (UTC)
replies: text: I mostly agree, except for point 2. The tensor product notation is important in differential geometry, as it is the manifestly invariant way of representing the quantities involved. I will make an attempt to clean up the section with clearer notation.Tazerenix (talk) 23:33, 9 February 2021 (UTC)
replies: text: I have made some effort to clean the article up, making everything into the same conventions (for example making sure all the tensor products T^*M \otimes E and E\otimes T^*M are in the right order, \operatorname{End}(E) = E^* \otimes E, and matching up these orderings with the indices of the local expressions). I also changed everything to inline math mode just to be consistent across the article (older sections were HTML and newer sections were LaTeX, and therefore I shamelessly changed the HTML to LaTeX, which I prefer). I also added some more modern general references.Tazerenix (talk) 02:51, 10 February 2021 (UTC)
thread_title:  Local expression 

message: I believe this article now has approximately all the mathematical content that should be presented on a wikipedia page for this topic. The following is a list of things that are needed before it would be considered in a good state:
* Better inline citations for the various parts of the content and where it appears in the general references appearing at the end of the article.
* A well-researched discussion of the history of the concept. At what point did the notion of a general connection separate from connections on the tangent bundle? Precisely who defined what, and in what references.
* An attempt at improving the exposition of the article to be more accomodating to the average reader. This task is partially meant to be absorbed by the connection (mathematics) article, so this is not necessarily the most pressing issue, however it may be worth going over the lead of this article.
* A brief discussion of the historical and contemporary research using connections on vector bundles, although this is mostly absorbed by the gauge theory (mathematics) article.
* A brief discussion of connections on complex vector bundles (i.e. (0,1)-connections, which are intimately related to holomorphic vector bundles), as well as potentially a discussion of notions of connections which appear outside differential geometry, for example in the context of algebraic geometry and over finite fields (local systems and so on are meant to be analogues of flat connections). These things could be better suited to other articles
* Including some more good diagrams, for example one could try to create pictures that help guide understanding of the curvature tensor, local frames and forms, the frame bundle, and holonomy. This is a geometry page after all, so we should endevour to include as many good pictures as possible.
* The structure of the various sections could be improved somewhat, perhaps in the style of the current incarnation of the metric connection article, where the different notational styles are clearly delineated, or by more liberally using subsections to split up content into bite-sized chunks.
These tasks would move the article from being a fairly comprehensive and clear reference for the basic theory of connections on vector bundles (as it is now), towards a more balanced description of the topic.Tazerenix (talk) 04:06, 10 February 2021 (UTC)
thread_title:  Tasks left