Connected holonomy is lower semicontinuous

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.