Connected holonomy is lower semicontinuous

TL;DR

This paper investigates the continuity properties of holonomy maps on manifolds, establishing that the restricted holonomy map is lower semicontinuous with respect to the $C^1$ topology on metrics.

Contribution

It proves that the restricted holonomy map is lower semicontinuous in the $C^1$ topology, advancing understanding of holonomy map stability.

Findings

$ ext{Hol}^0$ is lower semicontinuous w.r.t. $C^1$ topology

Continuity properties of holonomy maps are characterized

Results contribute to geometric analysis of metric stability

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.