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I suppose the following result follows from Ambrose-Singer theorem, but I cannot find a reference, and the arguments I found in the literature are usually weaker. The idea is that holonomy over a null-homotopic loop is bounded by the supremum of the curvature times the area of the 2-dimensional surface segment bounded by the loop.

THEOREM-CONJECTURE
Let $D$ be a unit disk, and $(B, \nabla)$ a trivial vector bundle on $D$ with connection (not necessarily orthogonal). Assume that the curvature $R$ of $\nabla$ is uniformly bounded, that is, $R(x, y)$ belongs to a compact subset $K$ in $End(B_m)$ for all $x, y \in T_m D$ of length 1. Then the holonomy of $\nabla$ around the boundary of $D$ is bounded by a uniform constant which depends on $K$ only.

I think I can prove this, but there are some segments of the proof which are tricky and take too much effort.

Can someone please point me to a reference, or to some relevant papers. Many thanks in advance.

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  • $\begingroup$ @user127309, any chance you would undelete your answer? I did not know about the Buser-Karcher reference. And your references are more precise than mine. $\endgroup$
    – Deane Yang
    Commented Jun 3, 2022 at 17:57
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    $\begingroup$ Done. Apologies, I did not see your comment. $\endgroup$
    – user127309
    Commented Jun 3, 2022 at 18:01

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There are in fact more precise versions, expressing the parallel translation around a loop as the identity map plus a curvature integral over a homotopy. References:

Section 3.1 of Werner Ballmann's lecture notes on vector bundles: http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/conncurv1999.pdf

Deane Yang's notes "Holonomy equals curvature" on his web page https://cims.nyu.edu/~yangd/papers/holonomy.pdf

Buser-Karcher, Gromov's almost flat manifolds, page 92.

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