Talk:Holonomy

Local holonomy

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)[reply]

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)[reply]

Illustration

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)[reply]

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!

When they say "tangent" they mean "tangent to the sphere" and not "tangent to the direction of motion". 67.198.37.16 (talk) 21:37, 7 November 2020 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

Yeah. Sorry. And no mention of the Aharonov–Bohm effect which is the holonomy on the U(1) circle bundle. Which is maybe the #1 reason why the holonomy is interesting to physics, and gets it's foot in the door. Oh and also the neutron interferometer, which, if I recall correctly and have not crossed any wires, is the the holonomy in a gravitational field (just parallel-transport the frame-field around a loop. Nothing happens to a scalar particle, but for a spinor, the phase twists around.) I'm thinking most interferometric effects in physics are holonomies, e.g. the Sagnac effect is a holonomy around the frame bundle in Minkowski space. Something like that. Shame, because those who understand this stuff throw these words out like magic spells with apocryphal origins in long-forgotten papers that themselves were never very clear. Fixing all this would be a very worthwhile and interesting project, as it can open the door to physics students who are otherwise overwhelmed by the raw weight and heaviness of the pure-maths approach. 67.198.37.16 (talk) 21:33, 7 November 2020 (UTC)[reply]

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parallel transport map $P_{\gamma }$

in the definition of "holonomy of a connection in a vector bundle", it states that $P_{\gamma }$ is a map taking closed loop to ${\text{GL}}(E_{x})$ , with $x$ being the endpoints of the curve $\gamma$ . The article immediately goes on to describe how the holonomy group at a point $y$ can be obtained from that of $x$ by the map $P_{\gamma }{\text{Hol}}_{x}P_{\gamma }^{-1}$ where $\gamma (0)=x,\,\gamma (1)=y$ . However, the article only defines $P_{\gamma }$ for closed loops. Of course it is relatively clear how to extend the definition to open loops, it just seems like the terminology can be introduced/defined more carefully. 128.178.172.90 (talk) 10:08, 24 January 2023 (UTC)[reply]

Local holonomy

Illustration

Any chance of an explanation of what holonomy is

External links modified

parallel transport map P γ {\displaystyle P_{\gamma }}

parallel transport map $P_{\gamma }$