Talk:Connection (mathematics)
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the second paragraph of general concept is unclear. What is B' and which bundle is induced etc. --MarSch 17:06, 19 October 2005 (UTC)
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[edit] 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.
[edit] General concept
The general concept can be summarized as follows: given a fiber bundle
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 
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 
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.
/Ends
[edit] 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)
[edit] 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.
