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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 28th 2010

added a stub entry for holonomy.

Just the bare definition, and of that even only the most naive one. Don’t have time for more. But created it anyway because I needed the link.

(Sounds a bit like like: I was young and needed the money…)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 28th 2010

okay, added an Idea-paragraph on the Ambrose-Singer theorem, too, but now I really need to look into something else.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeJul 28th 2010

I was thinking to add the Ambrose-Singer on the first look and than drifted to another work :) For Russian readers there is a nice proof in Postnikov, semester 4 of Lectures on geometry. I gave two student seminars in a Seminar on differential geometry in Zagreb, on that proof in Spring 1990, I think. Now I don’t remember the proof. As I grow older I am more and more stupid.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJul 29th 2010
• (edited Jul 29th 2010)

Wait a second, I am a bit confused with the entry now, section on Ambrose-Singer. I mean the parallel transport along infinitesimal squares does give a curvature that is a separate issue, but the Ambrose-Singer theorem says simply that the curvature evaluated at all pairs of horizontal vectors spans the Lie algebra of the group of holonomy. Parallel transport is needed to defined the holonomy group but not to define the spanning set for which only curvature is enough. Separately from Ambrose-Singer one then explains that curvature yes, is about local parallel transport. Now it is a bit mixed and I do not know how to streamline your exposition.

One detail: to get tangent vectors to holonomy group at point $p_0$ one takes all pairs of horizontal vectors $A,B$ at all points $p$ and then takes curvature form evaluated at $A,B$. The curvature form is $gl(n)$-valued, so one can take the span of all such results and this gives the holonomy Lie algebra. Maybe Urs is trying to directly relate this to the original invariant vector fields on holonomy group, so he transfers this result via Ad…but I am not quite sure.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 29th 2010
• (edited Jul 29th 2010)

it seems to me that I described the AS-theorem for points in base space, whereas you are thinking of the version formulated in terms of points in the total space of the principal bundle.

For points in the total space, the condition is that they need to be joined to the base point by a horizontal curve. That condition secretly encodes the Ad-action that I mentioned, speaking of paths in the base space. For if you pick a point not joined by a horizontal curve in the total space, the curvature 2-form there differs from the one at corresponding point in the same fiber which is horizontally joined by the Ad-action of the group element that takes one to the other.

Maybe later this week I find time to write more on this…

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 29th 2010

I should dig out the canonical references for this when I have time, but just quickly to address the above point: the version the way i stated it is for instance on page 32 here. Equation (4.2) defines that Ad-action, which then enters theorem 4.4.2.

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeJul 29th 2010

I added another proposition, on reduction.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 29th 2010

Thanks, Zoran.

I am hoping eventually we get a nice “topic-cluster” Connections and parallel transport which would then deserve to get its own floating TOC. At the moment the relevant entries are still too stubby, and I don’t feel I have much time for this. But eventually we should get there.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeJul 29th 2010

I hope the p-connection fits there as well. For now the top page connection is a good disambiguation page. Take some rest.

Sometimes people say word holonomy for parallel transport along not necessarily close path, though more rarely.