Talk:Connection (vector bundle)

Material from covariant derivative not yet incorporated

• One advantage of defining the connection in this way is that the following theorem becomes a statement about derivations on Lie algebras, to which one may then apply purely algebraic techniques from Lie algebra cohomology:

There exists a local trivialization of the bundle E with a basis of parallel sections if, and only if, the curvature vanishes identically.

• If P(E) → M is the frame bundle, consisting of all bases of E under the action of G = GL(r) where r is the rank of E, then an Ehresmann connection on P(E) induces a Koszul connection on E as follows. There is a natural one-to-one correspondence between (local) sections of E and functions φ : P(E) → R^r (defined locally over the base) which are equivariant under the action of G. Let the function φ associated to V be denoted f(V) for each local section V of E. The associated Koszul connection may be defined by

$f(\nabla _{X}V)=L_{X^{Hor}}[f(V)]$

where X^Hor is the horizontal lift of the vector field X, and L is the Lie derivative in the total space of P(E). (Included here. Silly rabbit 18:30, 31 May 2007 (UTC))

• If a connection in E (or a principal bundle associated with E) is specified by means of a parallel translation along curves, then a Koszul connection can be identified with the derivative of parallel translation. Let x_t be a curve in M, and let

$\tau _{0}^{t}:E_{x_{t}}\rightarrow E_{x_{0}}$

be the parallel translation in the fibres. Then

$\nabla _{{\dot {x}}_{0}}V=\lim _{t\rightarrow 0}{\frac {\tau _{0}^{t}(V_{x_{t}})-V_{x_{0}}}{t}}$

defines the associated Koszul connection. (Included in parallel transport article. Silly rabbit 18:30, 31 May 2007 (UTC))

(This material needs some work before it is suitable for inclusion here.) Geometry guy 10:58, 16 February 2007 (UTC)

Actually, your second point above is already included. See the properties section of the article. -- Fropuff 17:44, 16 February 2007 (UTC)

Very good. I just cut and paste this stuff in to make sure it didn't get lost (accidently - it may not be worth including) in the new structure. Anyway, I think it would be nice to draw out the relation with principal connections (in both directions) more visibly at some point. Geometry guy 00:55, 18 February 2007 (UTC)

Cf also the following from connection form.

• The connection form for the vector bundle is the form on the total space of the associated principal bundle, but it can also be completely described by the following form (on the base in a not invariant way). This subsection can be considered as a smoother but somewhat inaccurate introduction to connection forms...

• If one chooses a local trivialization of the vector bundle and takes $\nabla '$ to be the corresponding trivial connection, then $\omega$ gives a complete local description of $\nabla$.

• The choice of trivialization is equivalent to choosing frames in each fiber; this explains the reason for the name method of moving frames. Let us choose (a local smooth section of) basis frames $e_{i}$ in fibers. Then the matrix of 1-forms $\omega =\omega _{i}^{j}$ is defined by the following identity:

$\nabla _{u}e_{i}=\sum _{j}\omega _{i}^{j}(u)e_{j}.$

If $G\subset GL(F)$ is the structure group of the vector bundle and the connection $\nabla$ respects the group structure then the form $\omega$ is a 1-form with values in $g$, the Lie algebra of $G$.

Geometry guy 11:06, 16 February 2007 (UTC)

curvature - change in parallel transport

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)

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)

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)

Local expression

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)

Organization of connection-related material

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,π_TM/0,M).
• The corresponding nonlinear covariant derivative on a smooth manifold M into the direction of X∈TM/0 should refer to the differentiation D_X in the C^∞(M)-module of smooth vector fields on M determined by

$D:(TM\setminus 0)\times \Gamma (TM)\to TM\quad ;\quad D_{X}Y:=(\kappa \circ j)(Y_{*}X),$

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

$\nabla :TM\times \Gamma (E)\to \Gamma (E)\quad ;\quad \nabla _{X}v:=\kappa (v_{*}X),$

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)

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)

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)

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)

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)

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)

Assessment comment

The comment(s) below were originally left at Talk:Connection (vector bundle)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This article enriches the connections category but needs refinement. Geometry guy 22:44, 29 March 2007 (UTC)

Last edited at 21:43, 14 April 2007 (UTC). Substituted at 01:55, 5 May 2016 (UTC)

Nabla vs. D

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)

More notational edits required.

OK, I just finished a huge and quite unplanned expansion of the article on the metric connection. Upon completion, I realize that much or most of that content should be moved here. Here's what the problem is. This article is very nice and all, and it does successfully define the connection. But it is narrow: there are at least four different notational styles in use for the connection. I added these, and demonstrated, as best I could, that they really are just notational differences, and nothing more: these are all in the article on the metric connection. They really belong here, because essentially none of them actually require the metric connection; they are generically true, and thus belong here. I'm too exhausted to try this soon, but it really should be done. 67.198.37.16 (talk) 08:22, 24 October 2016 (UTC)

I plan to merge much of the content from metric connection into here, maybe sometime in the next few weeks. Prior or during that merge, I will rename all nablas to D's and all omegas to A's, leaving behind only a note explaining that nabla is a valid alternate notation. Should you be watching this talk page, and object to this, let's hammer out a solution that can work. 67.198.37.16 (talk) 17:12, 24 October 2016 (UTC)

@67.198.37.16: I have now added all of the content from the metric connection article that does not specifically pertain to the metric compatibility of the connection to this article (including the different ways of expressing the connection and curvature). In addition, the part of that article about Yang--Mills connections is now contained in the article Yang–Mills equations, so that article could now be simplified dramatically with the majority of that content being removed, and only a cursory discussion left with a link to this article instead.

Ideally the notational and mathematical conventions would be preserved between articles in the differential geometry wikipedia, and (shamelessly) since I have now standardised the notation in this article with Gauge theory (mathematics), when the metric connection article is updated the notation should be changed to match these articles. In particular this means using $\nabla$ instead of $D$ and using the convention $\Omega ^{p}(E)=\Gamma (\bigwedge ^{p}T^{*}M\otimes E)$ and $\operatorname {End} (E)=E^{*}\otimes E$. The latter are useful to keep track of throughout the various articles, because then the summation conventions used match up (in the end, these conventions should be kept consistent throughout the entire differential geometry wikipedia including the Riemannian geometry articles, but that is a much larger task to complete). As for using $\nabla$ v.s. $D$, this is a matter of convention but in my experience in the gauge theory literature $D$ is not particularly common. In fact the main convention is $A$ but this is a confusing abuse of notation for anyone learning the subject. There is a summary of the different conventions here.Tazerenix (talk) 03:47, 10 February 2021 (UTC)

Errata

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)

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)

Local expression

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)

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)

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)

Tasks left

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)