Connected holonomy is lower semicontinuous

TL;DR

The paper investigates continuity properties of the holonomy maps $Hol$ and $Hol^0$ on the space of Riemannian metrics ${\rm Met}(M)$, embedding them into the contexts of conjugacy classes and compact-connected subgroups via $PC_n$. A central contribution is proving that $Hol^0$ is lower semicontinuous with respect to the $C^1$ topology on ${\rm Met}(M)$ when restricted to $C^2$ metrics, under a uniform intrinsic-diameter bound on nearby holonomies. The authors also present a quantified Montgomery–Zippin theorem and a converse version, along with auxiliary lemmas on convexity radii and diameter control in homogeneous spaces, and utilize Wilkins’ loop-length bound to connect diameter to semicontinuity. Collectively, these results establish stability of the connected holonomy under metric perturbations and delineate the limits of semicontinuity phenomena for more general holonomy settings.

Abstract

In this article, we examine continuity properties of the maps $|Hol$ and $\Hol^0$ assigning, on a fixed manifold $M$, to a metric on $M$ its holonomy class resp. restricted holonomy class (conjugacy class of the connected component of the holonomy representation). Among related results, we show that $\Hol^0$ is lower semicontinuous w.r.t. $C^1$ topology on the space of $C^2$ metrics.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: Converse Montgomery-Zippin
• Theorem 6
• Theorem 7
• Theorem 8: Quantified Montgomery-Zippin Theorem
• Theorem 9
• Lemma 1