Talk:Connection (mathematics)

Unclear paragraph
the second paragraph of general concept is unclear. What is B' and which bundle is induced etc. --MarSch 17:06, 19 October 2005 (UTC)

Deleted from article
The following General concept section has been removed from the article. It fails its task in at least two ways: neither is it very general, nor is it much of a concept. In regard to its lack of generality, it manages to have abstracted away some of the most important characteristics of connections vis-a-vis some underlying structure on the base manifold (for instance, a metric, a projective structure, a conformal structure, a contact structure, an almost complex structure, etc.) For example, torsion is nonsensical from the point of view of the general fibre bundle Ehresmann theory of connections. Yes, there are ways of describing torsion, but they have an artificial feel to them: one needs a solder form or some other such device. Torsion is introduced as an ugly vestigial appendix to the connection formalism, rather than another invariant which, like curvature, occurs naturally. This is an inherent defect of the "Ehresmann" point of view of connections.

The second failure is that it is not much of a concept either. I much prefer the less rigorous transporting geometric data along curves approach to describing the notion of a connection. Firstly, there is no requirement a priori that the data must live in some fibre bundle. Although generally they do dwell in a fibre bundle of some kind, this fibre bundle is not usually so generic as to fit well with this highly abstract point of view. What, pray tell, happened to the view of a connection as a differential operator? as a one-form? as a class of frames with a rule for developability into a homogeneous space? What about this mysterious notion of "covariance"? These cannot all be neatly swept under the carpet of fibre bundles without raising some rather serious objections -- by myself, an expert, and from readers who may be meeting the idea of connection for the first time, or from some other point of view entirely.

Anyway, below I have preserved the contents of the offending section for posterity, at least until I figure out how to go about creating a suitable "General Concept" by way of introduction. Silly rabbit 08:30, 4 July 2006 (UTC)

The deleted paragraph
The general concept can be summarized as follows: given a fiber bundle


 * $$\eta:E\to B,$$

with E the total space and B the base space, the tangent space at any point of E has a canonical "vertical" subspace, the subspace tangent to the fiber. The connection fixes a choice of "horizontal" subspace at each point of E so that the tangent space of E is a direct sum of vertical and horizontal subspaces. Usually more requirements are imposed on the choice of "horizontal" subspaces, but they depend on the type of the bundle. (See Ehresmann connection for details.)

Given a $$B'\to B$$ the induced bundle has an induced connection. If $$B'=I$$ is a segment then the connection on B gives a trivialization on the induced bundle over I, i.e. a choice of smooth one-parameter family of isomorphisms between the fibers over I. This family is called parallel displacement along the curve $$I\to B $$ and it gives an equivalent description of connection (which in case of Levi-Civita connection on a Riemannian manifold is called parallel transport and for more general types of connections is sometimes referred to as development along a curve).

There are many ways to describe a connection; in one particular approach, a connection can be locally described as a matrix of 1-forms on the base space which is the multiplant of the difference between the covariant derivative and the ordinary partial derivative in a coordinate chart. That is, partial derivatives are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms.

On 'general concept'
Can I put in a plea for some extra talk about how partial differentiation is not 'available', as a geometric concept, though? This is pretty important, I feel, what with PDEs being the standard method of mathematical physics down the ages. --Charles Matthews 08:35, 4 July 2006 (UTC)


 * There is some discussion of that already over in covariant derivative. Should we expand that, and point to it from this article?  Or should we include a separate discussion here?  Silly rabbit 12:05, 4 July 2006 (UTC)

In that case it is probably kindest to amplify what is at covariant derivative. --Charles Matthews 17:58, 4 July 2006 (UTC)

Christoffel symbols
would you please clarify how the christoffel symbols involve no derivatives on u and v, but involve both the first and second derivatives of the transformation? without such an explanation, that part of the discussion sounds like gobbledygook to put it mildly.

“Connection”
Why is the word “connection” used? Is it implying that points in a space are connected in a particular way? —Ben FrantzDale 05:12, 27 January 2007 (UTC)


 * My understanding is that the word comes from the idea that it provides a way of connecting (or identifying) frames (infinitesimal coordinate systems on a manifold) at nearby points (infinitesimally close, or joined by a curve); the terminology was introduced by Cartan (see Cartan connection). These days the meaning is often understood in terms of Ehresmann connections on fiber bundles, where parallel transport (defined using horizontal subspaces) is used to connect nearby fibers. Geometry guy 13:55, 13 April 2007 (UTC)

Reworking the connections category
Fropuff and I have been putting in some work reworking and reorganising articles in the connections category, for which this is the main article. There are new articles on: Other articles such as covariant derivative and Ehresmann connection have been updated or rewritten in the light of this. Affine connection and Cartan connection have both been substantially rewritten. The material in connection form is now mostly redundant, and I have made a suggestion to use this article instead to make the link between the principal/fiber bundle and gauge theory points of view on connections and connection 1-forms, but that requires a lot of work.
 * Connection (vector bundle) (to which Koszul connection now redirects)
 * Connection (principal bundle) (which still requires quite a bit of work).

This talk page seems a good place to discuss the changes and the impact on this article. As this is the main article for the category, it should, in my opinion, provide as elementary an overview as possible of the concept of a "connection" and its history, both from the analytic (covariant derivative) and geometric points of view, introduce different notions of connection, and give some idea how they are related. There is already a lot of this here, but I find the article too technical and detailed in places - in particular, working out how the derivative changes under an explicit coordinate transformation is a nice idea, but probably too heavy for an overview article. Silly rabbit has just now been making some interesting changes to add a bit more geometry to the introduction. I will probably start to make some changes as well. Comments and suggestions welcome. Geometry guy 14:23, 13 April 2007 (UTC)

Intuition
I think I'm starting to understand this topic. Does this sound right?:
 * A connection relates the tangent space at a point on a manifold to the tangent spaces of nearby points. So if I am at point x and am moving in direction y (where y is necessarily in the tangent space of x), when I move a little bit, I'll be at point $$x'$$ and the connection tells me what direction, y', in the tangent space of $$x'$$ I am headed in.

Is that right? —Ben FrantzDale 03:19, 17 April 2007 (UTC)


 * Yes and no. It's true that a connection (or one sort of connection &mdash; a covariant derivative) relates the tangent space at one point to the tangent space at nearby points.  Also, the connection will tell you what direction you are headed in the new tangent space.  But it tells you even more: it tells you how other directions are related at point x and x&prime;, not just the one you're moving in.  The south pointing chariot gives a physical realization of a connection on the sphere.  It is a mechanical pointing device affixed to a chariot which uses a gear system to ensure that the pointer always points in the same direction (e.g., South), no matter how the chariot is moved.  If I move the chariot from x to x&prime;, it will then point in a direction South&prime; by keeping its pointer parallel throughout the motion (parallel to itself, not to the direction of motion).


 * One important subtlety is that South&prime; may not be true South, and this deviation from true South is more exaggerated the wider the path taken. Imagine the following alternative situation.  Instead of moving on the sphere, the chariot moves on a piece of graph paper, ruled by East-West and North-South lines.  In this case, the chariot really will point South no matter what.  But, if we try to wrap the graph paper around the sphere so that the East-West lines are latitudinal and the North-South lines are longitudinal, we find that it's impossible.  Longitudes get more tightly packed towards the poles of the sphere: the sphere is curved.  So if the south pointing chariot were to confine its movement to the equator then the pointer will always point true South.  But once it moves off the equator, its pointer will deviate increasingly from true South.


 * Would it be helpful to include a (cleaned up) explanation such as this in the article? Silly rabbit 09:03, 17 April 2007 (UTC)


 * Thanks for the reply. That is consistent with what I had in mind. This article could do with some additional diagrams. I may give that a shot once my semester is over. —Ben FrantzDale 10:35, 17 April 2007 (UTC)

Family tree of connections
Wouldn't be helpful to include in the article the "family tree" of the different kind of connections and some attached notions?

I mean something like this:

connection
 * Cartan (Connection form)
 * Projective
 * Affine (parallel transport)
 * Ehresmann
 * Principal
 * Koszul (Covariant derivative)
 * Metric
 * Levi-Civita (Christoffel symbols)
 * Grothendieck
 * Gauss-Manin —Preceding unsigned comment added by 86.101.4.111 (talk) 09:30, 7 October 2007 (UTC)

=Peanut gallery
Good article. I am what you should term a layman. I hadn't heard of connections before today. I stumbled across them in reading Wiki on General Relativity and the various ways to describe the particle and event horizons. This is about as deep as I want to get into the math, now. Penrose's Road to Reality discusses some of the same math slightly differently, FYI. That is, there is life outside of Mathematics and this stuff does mingle there. Does the article mention that? But your comments above that both the novice and more advanced readers should get something out of this is a good point. Be kind to us concrete thinkers. Going too abstract would really narrow the readership. —Preceding unsigned comment added by 69.40.243.166 (talk) 01:39, 3 February 2008 (UTC)


 * Thank you for taking an interest. It seems that a better article for your particular set of interests might be Introduction to the mathematics of general relativity.  I will add a link here. Silly rabbit (talk) 14:10, 8 February 2008 (UTC)

Teleparallelism and affine connections
An editor, User:RQG, seems quite keen on introducing material on Teleparallelism, Einstein-Cartan theory, and affine connections into the lead paragraph of the article. I oppose this for a number of reasons.

This article is about the general notion of connection in mathematics. Introducing specializations of this concept, such as affine connections, in the first sentence does not seem consistent with this goal. There is a variety of related notions of connection to choose from: it isn't just affine versus teleparallel. There are Cartan connections, principal connections, connections in a vector bundle of which covariant derivatives and affine connections are a special case, Ehresmann connections in a fibre bundle, and the (perhaps most general of all) Grothendieck connections.

Somehow the first sentence needs to communicate the general features common to all of these. Connections in this general sense have little to do with transporting "coordinate systems." Rather one is transporting a general kind of geometrical data along curves. For instance, Cartan connections transport a generalized contact with a homogeneous space along curves in a manifold. While it is true that, in a sense, most connections of importance in geometry have an interpretation which involves the transport of frames, this is by no means what the connection "is", nor is it legitimate to equate frames and coordinate systems.

I would like to point out that, apparently contrary to the proposed change, teleparallel connections are (a special class of) affine connections. They clearly don't belong in the first paragraph of the article, given their relatively minor importance in the general theory of connections.

Finally, of course any change is welcome which makes the idea clearer, even if it uses a special case of the general concept. But a misleading and confusing addition whose sole purpose seems to be to promote teleparallel connections does not improve the article. silly rabbit (  talk  ) 12:11, 21 March 2008 (UTC)

A connection is not just an affine connection
silly rabbit seems to wish to reduce the notion of a connection to parallel transport on a curve, which is in fact just an application of a particular type of connection. As described in earlier discussions here, and in other wikipedia articles, the word "connection" refers to a connection between tangent spaces, or between coordinate systems. Equivalently, and perhaps more easily defined and understood, it gives a definition of parallel between vectors, including the vectors used to define coordinate axes. This is a general article on connections, and it should not limit the concept by removing specific types of connection or reducing the notion of connection to a specific application. User:RQG 10:00, 24 March 2008 (GMT).


 * No, the description given in the article is a perfectly general one, applicable to any sort of connection. Please find a version of a connection which does not involve some notion of parallel transport.  Even Ehresmann connections have parallel transport maps associated with them.  I have made a few changes to the lead aimed at simplifying the first paragraph to make it more readable, but I certainly do not think that every connection is affine, or any other such nonsense.  silly rabbit  (  talk  ) 10:40, 24 March 2008 (UTC)

The current version is extremely muddled. It both defines a connection as being along a curve, and defines an affine connection as being along a curve. When Einstein introduced the remote connection, the purpose was to get away from that. Of course it is true that one can regard a connection defined between tangent spaces at A and B as making it possible to transport vectors from A to B, but that invokes an additional concept, one of transport, whereas a connection need only invoke the notion of parallel. I think you should learn what a connection is, rather than borrow simplistic ideas from badly written textbooks, which is how the article is currently written. —Preceding unsigned comment added by RQG (talk • contribs) 12:47, 24 March 2008 (UTC)


 * Yes. A connection is a way of moving data along a curve.  An affine connection says "The data being transported consists of tangent vectors."  All connections, regardless of their origin, can be defined in terms of transport maps along curves.  This is one reason that the present article settled on this: it is universal enough to be applied to any one of the available definitions of connection.  Including those of Einstein and Cartan.  And those of Ehresmann.  And those of Koszul.  And those of Shiing-Shen Chern.  Also, I'm not sure what you mean when you say that a connection need only invoke the notion of parallel.  Please elaborate (here on the talk page).  silly rabbit  (  talk  ) 13:04, 24 March 2008 (UTC)

A connection defines the notion of parallel between vector spaces defined at two points of a manifold. An affine connection is defined for points at infinitesimal separation, and in that case one may extend the notion to parallel transport along a curve, but this follows from the basic definition, and should not be a part of it because definitions in mathematics should be as simple as possible. It is perfectly possible to define a manifold consisting of disconnected regions, which are not connected by a curve. It is also possible to define a discrete manifold, in which curves make no sense. This was refered to by Riemann, though I haven't seen it studied. It is also undesirable to make curves essential to the definition because the notion of a single path in quantum theory is suspect, at the very least. In the case of a teleparallel connection, parallel transport is not path dependent, so even if one can define a curve between remote points, the fact of doing so is irrelevant to the definition. —Preceding unsigned comment added by RQG (talk • contribs) 17:58, 24 March 2008 (UTC)


 * Teleparallelism is a very special case of the general affine connection concept; one in which the curvature vanishes identically (thus by the Ambrose-Singer theorem, the parallel transport is not path dependent). Whether one chooses to introduce things infinitesimally or in terms of curves is largely a matter of taste.  However, I have never seen a particularly compelling introduction to connection theory using the infinitesimal approach which is understandable by all persons who might have use for an encyclopedia article on the subject.  I highly doubt it is possible to write a simple description of the infinitesimal approach to connections, and so far you haven't provided one.  (A simple description which covers connections in more general fibre bundles, not just vector bundles.)   silly rabbit  (  talk  ) 19:03, 24 March 2008 (UTC)

A moment ago you said not every connection is affine, now you say it is. If the connection is defineable on any curve it is affine. A teleparallel connection does not imply curvature vanishes. I have given instances where one may want to define a connection which is not defineable in terms of a curve. The infinitesimal approach to connections is specific to affine connections. It would be better to explain the general notion of a connection, not one which applies in a subclass. —Preceding unsigned comment added by RQG (talk • contribs) 22:39, 24 March 2008 (UTC)


 * I never said (nor implied) that every connection is affine. If you believe that I have, then I think you should probably go an see what an affine connection really is.  Parallel transport is defined for any sort of connection: see the articles connection (principal bundle), Ehresmann connection, Cartan connection, Covariant derivative, parallel transport, holonomy, affine connection, and connection (vector bundle) for examples.  This is parallel translation of the fibre defined along a curve in the base manifold.  Symbolically, if F &rarr; M is a fibre bundle and &gamma; : [0,1] &rarr; M is a smooth curve, then there is a map &Gamma;t defined for $$0\le t\le 1$$ such that $$\Gamma_t : F_{\gamma(0)} \to F_{\gamma(t)}$$ maps the fibre of F over the initial point of the curve to the fibre over its final point, satisfying some obvious axioms (details are in the parallel transport article).  These maps can be given for any reasonable definition of connection in a fibre bundle.  Let me again emphasize, this is true for a connection in a fibre bundle, not just the tangent bundle (which would give an affine connection, provided the &gamma; maps are linear), nor just a vector bundle (which would give a vector bundle connection, again provided the &Gamma; is linear), nor even just on a principal bundle (which would give a principal connection provided the &Gamma; is equivariant) but a full-blown fibre bundle.  This also has generalizations to a variety of other kinds of fibrations.    silly rabbit  (  talk  ) 23:29, 24 March 2008 (UTC)

The article on affine connection is perfectly correct, and makes the connection between nearby tangent spaces. It also states "A choice of affine connection is also equivalent to a notion of parallel transport". Thus by restricting the definition of connection to parallel transport on a curve, you are restricting to affine connections. A teleparallel connection is not necessarily affine. Consider the land areas of the surface of the earth. We can define a teleparallel connection by means of compass directions, equivalent to a submanifold of a Mercator projection. The connection is defined without reference either to a curve, or to transport, and curves cannot be defined between all points. Even on a full Mercator projection, defining the connection in terms of a curve would be extremely unnatural. —Preceding unsigned comment added by RQG (talk • contribs)


 * I'm sorry, but this is completely wrong. An affine connection is equivalent to giving a notion of parallel transport in the tangent bundle.  Every other sort of connection gives parallel transport in some other fibre bundle (vector bundle/principal bundle/whatever).  Please go read the articles linked to above.  Teleparallel connections, as I have already indicated, are nothing more than affine connections with vanishing curvature.  Sure, you can define them in terms of special coordinate systems, but this is a very special phenomenon that is specific to only this one kind of seldom studied type of connection.  Generally, flat connections are sort of special, and should be given special treatment rather than advanced as examples of what to expect from the typical case.   silly rabbit  (  talk  ) 01:41, 25 March 2008 (UTC)

I find it quite alarming that you confuse a connection with curvature. A connection should not be called flat. As originally pointed out by Cartan, it requires both a metric and a connection to specify curvature. A Mercator projection is not an esoteric case, and nor does the geometry described by a Mercator projection have vanishing curvature. —Preceding unsigned comment added by RQG (talk • contribs)
 * It is alarming that you don't know that "flat connection" is standard terminology. --Lambiam 03:03, 26 March 2008 (UTC)


 * Curvature is an invariant of the connection. This is a routine mathematical fact.  Connections with vanishing curvature are called "flat."  I'm sorry if this is new or alarming to you, but it is completely standard in mathematics.  I don't see what the Mercator projection has to do with anything.   silly rabbit  (  talk  ) 11:27, 25 March 2008 (UTC)


 * FYI: A connection in the tangent bundle is an affine connection.  That is what it means to be an affine connection.  Please read up more about this at affine connection.   silly rabbit  (  talk  ) 10:41, 24 March 2008 (UTC)


 * Note: The word affine means close or nearby, and has the same root as affinity. It is logically opposite to a remote connection. It is not true that any connection in the tangent bundle is made between nearby tangent spaces. —Preceding unsigned comment added by RQG (talk • contribs)


 * I'm not sure what you are trying to argue here. The term "affine connection" was coined in the 1930s by Elie Cartan as a geometry modeled on affine space (a term already in common mathematical usage at the time).  Sharing a common root with affinity, while somewhat provocative, is largely accidental.  I really don't understand your last sentence.  Could you give an example of a connection in the tangent bundle (which I presume to mean a linear connection) but for which "nearby" tangent spaces are not connected?   silly rabbit  (  talk  )

I think you should read the article on affine connection. I have already given examples of a connections which are not defined specifically for nearby tangent spaces, in which parallel transport is not path dependent, and which there are points in the manifold which cannot be linked by a curve. The point of a remote connection is that it is not defined only for nearby tangent spaces, and for that reason is distinct from an affine connection. RQG (talk) 13:08, 25 March 2008 (UTC). The article linear connection) appears to be written largely by you and repeats basic errors, like saying that curvature is a property of the connection.


 * Ridiculous! I made a total of three edits to that article.  One of these was a self-revert, so I made a net of one edit!  This edit was a redirect made before there was even an article.  Neveretheless, that article looks fine to me.   silly rabbit  (  talk  ) 13:19, 25 March 2008 (UTC)

Honestly, I have better things to do than explain to an argumentative novice what an affine connection is and what it isn't. If you have something definite to bring to the discussion, then do so. Otherwise, I am not going to continue to argue. silly rabbit (  talk  ) 23:29, 24 March 2008 (UTC)


 * I hope then that you will withdraw your objection to the removal of transport on a curve as the definition of a connection. The article as it stands is wrong. Invoking connections on fibre bundles seems quite innappropriate for an encyclopedia article. —Preceding unsigned comment added by RQG (talk • contribs)


 * I will withdraw my objection when you offer a suitable alternative, which you have not done. By the way, why are fibre bundles inappropriate for an encyclopedia article?   silly rabbit  (  talk  ) 01:22, 25 March 2008 (UTC)

Because fibre bundles are mathematics jargon, and would restrict the readership to mathematicians. It is more likely that a layman, or a physicist or cartographer, wishes to look up a connection, seeking more detail and clarity than he is given in in a textbook. —Preceding unsigned comment added by RQG (talk • contribs)


 * And that's why there are articles like covariant derivative, connection (vector bundle), and so forth. There are bunches of connections out there: one for each type of data studied.  Honestly, the lead is not that bad as far as comprehensibility, in my opinion.  It is at least mathematically correct (despite your opinion), and it does not bring in the idea of a fibre bundle.  However, unlike your version, it does not exclude fibre bundles from consideration, and moreover does not overemphasize teleparallel connections which are studied far less than connections in fibre bundles.   silly rabbit  (  talk  ) 11:14, 25 March 2008 (UTC)


 * As you have it the lead remains incorrect RQG (talk) 13:08, 25 March 2008 (UTC)

I think that you two need to calm down. This discussion started off as a well mannered one, and in good faith. I am inclined to agree with Silly Rabbit's point of view, but I don't agree with his attitude. You are supposed to be nice to each other. Comments like "Honestly, I have better things to do than explain to an argumentative novice what an affine connection is and what it isn't." are not, to my eyes, nice. Silly rabbit is correct when he says that curvature is a property of a connection. In fact, for affine connections, the curvature tensor and the torsion tensor are two of the very first objects one meets after getting to grips with a connection. If I recall, the curvature tensor is given by
 * $$ K(X,Y) := \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]}, $$

where $$\nabla$$ is the connection in question, $$[-,-]$$ is the Lie bracket, and X and Y are vector fields over the manifold. The torsion tensor is given by
 * $$ T(X,Y):= \nabla_XY - \nabla_YX - [X,Y] . $$

If K(X,Y) is zero for all vector fields then $$\nabla$$ is called flat. This means that if we parallel transport a vector along any closed curve, when we arrive back at the start the vector will be the same as the original. This says that the local symmetry group is just the identity. If T(X,Y) is zero for all vector fields X and Y then the connection is called torsion free. The property of being torsion free doesn't have such a nice geometrical interpretation, but it is a nice property to have. On a Riemannian manifold there is a unique torsion free connection $$\nabla$$ where the matric g is such that $$\nabla g = 0$$. In affine differential geometry we play the same game, and there is a unique torsion free connection where the volume form is such that $$\nabla\omega = 0$$ (and another small condition is satisfied). Declan Davis (talk) 11:03, 17 September 2008 (UTC)

Comments

 * Comment. I do not conflate connections and affine connections.  RQG seems to think that any connection in a fibre bundle in which there is a transport map along a curve is an affine connection.  Thus he/she insists on a false dichotomy between affine connections and "teleparallel connections".  This is not the case.  Indeed, every sort of connection (even a "teleparallel connection") is equipped with some kind of parallel transport maps.  Please see the article parallel transport for an introduction, as well as the article Ehresmann connection for the general theory of parallel transport for a general connection.  See also connection (vector bundle), which emphasizes parallel transport in vector bundles, and holonomy in which parallel transport in a vector bundle or a principal bundle plays a decisive role in modern differential geometry.  Finally, emphasizing parallel transport as a unifying theme among the articles on connections has emerged as the result of a broad community consensus.  I and two other experts, User:Geometry guy and User:Fropuff, have all agreed to use parallel transport as the unifying theme for the connection articles.  Furthermore, nearly all of articles in the Category:Connection (mathematics) emphasize parallel transport, and have done so for a considerable amount of time.  Per WP:SILENCE, this tacitly implies a broad community consensus.   silly rabbit  (  talk  ) 14:08, 25 March 2008 (UTC)

I have presented instances where a teleparallel connection is not given by parallel transport on a curve, and it is easy to construct others, for example, teleporting as might be used in a computer game. The claim that all connections can be represented as parallel transport on a curve is thus false. Moreover, definitions in mathematics should be elemental where possible. Parallel transport consists of many applications of an affine connection and is not. If there is a policy to define connections in terms of parallel transport, this should be reversed. It fails to describe the full concept of a connection and is therefore inaccurate. RQG (talk) 14:52, 25 March 2008 (UTC)


 * Please give references on what you are regarding as a "teleparallel connection." In the article teleparallelism, the connection described most definitely is an affine connection.  If you feel that somehow the description given in the article is missing some critical feature of the relevant mathematical literature, then the burden is on you to give references supporting that thesis.  For my own part, I can personally assert that, among dozens of standard references from differential geometers and mathematical physicists emminent in their fields, the standard definition of a "connection" is completely compatible with what is written in the lead of the article here, and is discussed in most of the connectoin articles on Wikipedia.   silly rabbit  (  talk  ) 15:22, 25 March 2008 (UTC)


 * Some easily verifiable reliable sources are at the Springer Encyclopaedia of Mathematics. Here is their main article on connections, and here is their article on Connections on a manifold.  These two articles on the general notion of a connection in fact rigorously define connections in terms of parallel transport maps.   silly rabbit  (  talk  ) 16:04, 25 March 2008 (UTC)

I have already given examples. For another, use the definition of teleparallelism from Einstein's paper which you can get at http://www.lrz-muenchen.de/~aunzicker/einst.pdf, on a manifold consisting of two spheres. I have reasons to think this is important, even if it sounds odd. Those reasons do not matter to this discussion. The fact that it is possible is sufficient to refute the treatment based on parallel transport. —Preceding unsigned comment added by RQG (talk • contribs) 16:18, 25 March 2008 (UTC)


 * Einstein does not seem to discuss the case of two spheres. So this reference does not support your point.  Please find another.  silly rabbit  (  talk  ) 16:28, 25 March 2008 (UTC)

What Einstein actually does in the paper is irrelevant to the existence of such a manifold. RQG (talk) 17:26, 25 March 2008 (UTC)


 * Also, the only place in that paper where he uses the word "connection" it is in reference to the associated infinitesimal connection, which as I have alreay indicated is (in this case) an affine connection. Next!  silly rabbit

( talk  ) 16:31, 25 March 2008 (UTC)

Again, we are not concerned with what Einstein does, but with mathematical definition which is an objective concept. RQG (talk) 17:26, 25 March 2008 (UTC)


 * Right. So why don't you please give references which define a connection the way you would like it to be defined.  We may then discuss the merits of those references.  So far, the Einstein reference does not support your contention.  Do you have any other sources?   silly rabbit  (  talk  ) 17:33, 25 March 2008 (UTC)

This paper may be a little better for the purpose http://www.lrz-muenchen.de/~aunzicker/rep1.pdf, first para of section 2. The point is that parallel is defined directly between vectors at remote points P & Q, without reference to path. One can reduce this to an affine connection only in manifolds with suitable topology. I have given examples of manifolds with structure which does not allow such a definition to reduce to an affine connection. RQG (talk) 21:41, 25 March 2008 (UTC)


 * It is really unclear to me whether Einstein is regarding these as global objects or just local ones. He does use the word "connection" to discuss his parallelism, but he then (in the next few lines) goes on to derive the "connection" coefficients.  So it really is not at all a clear thing whether he means us to think of the connection as global (in the sense of your example of two spheres) or local.  Anyway, when I say "reference", I mean one like one of the ones I have given, where there is a definition of a connection of the following sort:  "A connection on a manifold consists of the following data..."  In a field as thoroughly established as the study of connections, it should be trivial to do this.  The fact that you are unable to produce anything (save a vague mention in one of Einstein's papers) to counter the unambiguous sources I have given (here on the talk page, as well as in the articles in the category) is quite telling.


 * Anyway, I have had some further thoughts on the matter. In this paper Einstein is talking about what is nowadays referred to as an absolute parallelism.  (Note: they are not normally called connections in, say, post-1950 mathematics.)  An absolute parallelism is, of course, different from an affine connection.  However, it always determines a flat affine connection, which is the one of interest to the teleparallelists (presumably).  Conversely, given a flat affine connection (on a connected and simply connected manifold), one can obtain an absolute parallelism by parallel translation.


 * Hence, I remain unconvinced that the article should be adjusted to include global path-independent notions of connections. Perhaps you should write an article on the topic first, and then try to edit this page rather than go about things the other way around.   silly rabbit  (  talk  ) 22:13, 25 March 2008 (UTC)

Okay, I withdraw. Thanks for the discussion, which I have found useful in the end. RQG (talk) 15:53, 26 March 2008 (UTC)

Comment All connections involve parallel transport, and this is a top level article treating the most general features of connections in mathematics. From reading the exchange, it is clear that RQG does not understand the article nor, it seems, the very notion of connection. Especially revealing in this regard is his/her comment about "teleporting in a computer game". Beam me up, Scotty! I see little point in arguing about "teleparallelism" (whatever that is) at this page. Silly Rabbit informed him of the broad consensus among the expert editors (reflecting the views of the mathematics community at large) and pointed out an easily accessible reference (Springer EOM). Go and learn! Arcfrk (talk) 23:32, 25 March 2008 (UTC)

No one suggests teleporting as a model of reality. Nonetheless, a computer game is a valid mathematical model - or should be if it is not bug free. Designing bug free programmes is an important field in its own right. RQG (talk) 15:53, 26 March 2008 (UTC)

Comment I second the comments by Silly Rabbit and Arcfrk. --ShanRen (talk) 20:31, 26 March 2008 (UTC)

Comment, Yep, Silly Rabbit and Arcfrk present the common, consensus view of the physics and mathematics community. WP needs to stick to that consensus view. linas (talk) 03:31, 27 March 2008 (UTC)

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