Talk:Connection (mathematics)

message: ===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)
replies: text: \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)

text: 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)
thread_title:  General concept 

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

message: 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)
replies: text: 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)
thread_title: On 'general concept'

message: 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)
replies: text: 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)
thread_title:  “Connection” 

message: 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:
* Connection (vector bundle) (to which Koszul connection now redirects)
* Connection (principal bundle) (which still requires quite a bit of work).
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.
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)
thread_title:  Reworking the connections category 

message: 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)
thread_title:  Family tree of connections 

message: 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)
replies: text: 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)
thread_title:  =Peanut gallery 

message: 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)
thread_title:  Teleparallelism and affine connections 

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