Talk:Holonomy

Jump to navigation Jump to search
WikiProject Mathematics (Rated B-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 High Importance
Field:  Geometry

The corrections by User:Serenus suggest a couple of things to me (I'm not an expert in these matters):

(1) Is there a necessary distinction to be made, between holonomy and local holonomy?

(2) Is this topic connected with the Berger list, which is a Requested Article?

Charles Matthews 10:22, 10 May 2004 (UTC)

Answering (2) myself, it seems clear from some Googling that it's 'yes' (for example http://arxiv.org/abs/dg-ga/9508014). But that recent work has shown up some gaps. So, redirecting Berger list here, and adding a note.

Charles Matthews 10:30, 10 May 2004 (UTC)

The illustration of holonomy on the sphere is wrong! The vector field on the left-hand meridian should be perpendicular to the great circle, rather than tangent.

Also, the distinction above should be between holonomy and REDUCED holonomy.

I believe you may be misreading the illustration. The holonomy is at the point A, not the point N (as your post seems to suggest). The "loop" in the diagram is perhaps misleading (since it suggests starting and finishing at N instead of A), and I will do my best to edit the image to move the loop into the correct position when I get the chance. But thanks for catching the mistake above (and in the article). You may have missed a few other instances, and I will fix the remaining cases where it was wrong. silly rabbit (talk) 12:20, 24 March 2008 (UTC)

There are still serious deficiencies in the illustration:

1. The three segments are apparently parts of great circles, and so geodesic. Hence the tangent vector to each must be parallel. But notice that the "transported" vector is drawn as tangent to the equator at B, but not at N!

2. For a triangle consisting of two lines of latitude and an equatorial segment, the holonomy angle alpha should equal the change in latitude -- i.e. the interior angle of the triangle at the vertex N. This also reflects the fact that, on a unit 2-sphere, the holonomy angle equals the area of the triangle (= integral of the Gauss curvature, which is +1 in this case). This

```could be clarified by indicating that the triangle is supposed to have three right angles, assuming you want alpha to be 90 degrees.
```

Comment: There seems to be some confusion about the illustration. To discuss the holonomy of a sphere, all direction vectors should be tangent to the sphere, for there is no other notion of direction available. However if one is considering holonomy for the curve in R^3 that happens to sit on the surface of the embedded sphere, then the holonomy should be zero and all the vectors should be "distantly parallel", the common notion for euclidean 3-space(the holonomy group of R^n =0), since parallel transport over the curve would be parallel transport in R^3 whether or not the sphere were present. Clearly one doesn't want the latter case for this illustration, so it should be cleaned up to make the vectors look like they are tangent to the surface. KYSide (talk) 03:37, 21 October 2011 (UTC)

3. The circular arrow, indicating the direction of transit, has a gap near N rather than at A. This is not good pedagogy, because it leads the reader believe that the trip starts at N rather than at A. —Preceding unsigned comment added by 71.167.177.39 (talk) 14:03, 29 October 2010 (UTC)

Any chance of an explanation of what holonomy is

The opening sentence is totally meaningless to anyone who doesn't already know what holonomy is.

And the article gets worse. Cannonmc (talk) 08:31, 29 January 2014 (UTC)

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Holonomy. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

As of February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template `{{sourcecheck}}` (last update: 15 July 2018).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot 01:39, 6 November 2017 (UTC)