Talk:Holonomy

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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[edit]

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)

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