Talk:Parallel transport

Generality of this article?

There are various levels of generality at which one can discuss parallel transport:

for vector fields;
for sections of arbitrary vector bundles;
for sections of arbitrary geometric bundles (e.g. frame bundles and their associated bundles);
for sections of arbitrary fiber bundles (e.g. principal bundles and their associated bundles);

At what level should this article employ most of its energies? Geometry guy 16:08, 12 March 2007 (UTC)[reply]

Why not all of them? But if I had to pick one, I believe it should focus primarily on vector fields since that is likely to be the area of interest to most people looking in the general community. At least, it should start out with vector fields. The other levels of generality are also important, though, for the article holonomy (which also needs a lot of work). Silly rabbit 14:11, 13 April 2007 (UTC)[reply]

I tend to agree: work with vector fields, then explain that it all generalizes to each of the above levels (with fewer details in the more general cases). Geometry guy 14:41, 13 April 2007 (UTC)[reply]

This is certainly the correct answer, but as of 2024 this vision is still yet to be realized. I attempted to put some work on it though. Mathwriter2718 (talk) 14:19, 3 July 2024 (UTC)[reply]

Another question

In the earlier part of the 20th century, connections were defined in terms of parallel transport satisfying certain properties (see e.g. the Springer Encyclopedia articles on connections). Should we present this approach here? Geometry guy 22:21, 20 March 2007 (UTC)[reply]

I'm all for it, since PT really is the unifying theme behind all the connection concepts. So organizing Category:connection (mathematics) around this idea was one of my original goals in this bulk revision. That said, I don't think it can be done in a way that will be at all familiar to readers who have just recently encountered the covariant derivative (*) and are looking for some further reading on the subject. So, if it is to be done, it shouldn't be the primary approach: perhaps a section later on in the article like Defining the covariant derivative in terms of parallel transport.

(*) Ok, anywhere except Kobayashi-Nomizu.

Silly rabbit 14:20, 13 April 2007 (UTC)[reply]

I agree with your conclusion here, especially as this is an encyclopedia, not a new book on connections ;) ! Also, I would not say PT is the unifying theme, since there are several points of view, such as Cartan's infinitesimal one. And parallel transport is quite hard to use as a primitive definition (see again the Encyclopedia of Mathematics) because one has to capture the idea that it is not arbitrary, but only depends on 1-jets. This involves taking an awkward derivative at time zero. I feel some of the infinitesimal intuition gets lost when PT is overemphasised. I hope with both of us working on this we can achieve a good balance between the different points of view ;) Geometry guy 14:41, 13 April 2007 (UTC)[reply]

For future reference, some starting material for this point of view is buried in Talk:Connection (vector bundle). Silly rabbit 19:23, 13 April 2007 (UTC)[reply]

Yes, I put it there, to save it from being buried even deeper in the history archives. Geometry guy 19:28, 13 April 2007 (UTC)[reply]

Transporting tensors on Lie groups

I'm trying to understand how in general one transports tensors in the tangent space of a point in a Lie group to some other point in a Lie group, particularly using matrix representation. For example, the exponential map on SO(3) lets you represent tangent vectors in SO(3) as skew-symmetric matrices, and there is a 1:1 mapping between 3x3 skew-symmetric matrices and 3-vectors, however they transform differently... If I have an axis-angle vector, $\omega$ , I can rotate it (transport it) using a rotation matrix: $R\omega$ . However, if I have the skew-symmetric representation $\omega$ , $[\omega ]_{\times }$ , I believe it transforms like a matrix; that is,

[R\omega ]_{\times }=R[\omega ]_{\times }R^{\top }

.

Two questions:

In SO(3), it seems like the "matrix" representation of a "second-order" tensor would be a (3x3)x(3x3) matrix. How would that transform? Presumably you'd need to transform both 3x3 portions, so it would look something like $R_{ij}R_{nk}T_{jk\ell m}R_{\ell p}R_{qm}$ , which seems pretty kludgy.
Outside of SO(3), the exponential map on a matrix gives you tangent "vectors" that are matrices with no obvious "simpler" representation like an axis angle. There must be a wholistic way of transporting tangent vectors and tensors. Is it simpler than constructing some high-order object and transforming along each index as above? —Ben FrantzDale (talk) 02:42, 5 November 2009 (UTC)[reply]

Equation for parallel transport in general relativity

It would be useful to include the equation for parallel transport of a vector as used in general relativity:

${\frac {dv^{\mu }}{ds}}+\Gamma _{\alpha \beta }^{\mu }{\frac {dx^{\alpha }}{ds}}v^{\beta }=0$

But where to put it in the article? 193.198.162.14 (talk) 09:49, 6 February 2018 (UTC)[reply]

Illustration of Parallel Transport

It appears that the vectors illustrating the parallel transport are viewed form a point above the equatorial plane. The vector at B is shown properly. The direction of the vector between A and B is not correct. It should be tangent to the equator, i.e. showing downward from the equator, just like the vector at B. Also, the vector at A showing to the N-S line is in error. It should be tangent to the equator, i.e. directed downward from the equator. It is absolutely misleading to end it at that line. It has nothing to do to the N-S line.178.48.163.158 (talk) 10:56, 20 August 2020 (UTC)lb[reply]

For whom is this written?

This article only raises questions. It explains parallel transport on a level only experts understand. It confirms a well-known fact that academics can't explain things, because they don't have any empathy.

Problems with this article

This article has some serious problems.

1. There is not even a single example.

2. It includes the case of a general vector bundle and even further generalizations beyond that (as it should), but it barely discusses the case of the tangent bundle. Most readers of this page probably only care about this case.

3. Then there is a section called "Parallel transport in Riemannian geometry", even though the previous section was also about Riemannian geometry.

I will try to make some first steps towards fixing these problems. Mathwriter2718 (talk) 13:06, 3 July 2024 (UTC)[reply]

I went ahead and removed the section "Parallel transport in Riemannian geometry" and added a new section "Parallel transport of tangent vectors" that I think is written for a more appropriate audience. This new section is spliced together from 1. other parts of this page that I moved, 2. content I recently wrote for Riemannian manifold, 3. the part about parallel transport from Affine connection. This section and the article as a whole still needs significant work. Mathwriter2718 (talk) 14:17, 3 July 2024 (UTC)[reply]

Looks good, except I think the definition of the linear parallel transport map

\Gamma _{s}^{t}

should be properly introduced in the tangent vectors section. (It is referred to in the Metric connections part before the introduction later in the vector bundles section.) Tito Omburo (talk) 15:38, 3 July 2024 (UTC)[reply]

Certainly true. I went ahead and did this. Mathwriter2718 (talk) 18:59, 3 July 2024 (UTC)[reply]

Wonderful! Tito Omburo (talk) 14:20, 4 July 2024 (UTC)[reply]