Lower semicontinuity of restricted holonomy groups

Abstract: A result on the monotony of holonomy groups of metric connections in limits is proven. The main outcome is that given a sequence of metric connections that are Lipschitz continuous and converging in $C<sup>0$, if the connections in the sequence all have holonomy contained in a closed group $H$, then also the limit connection has holonomy contained in $H$. In particular, this finds an application for Riemannian holonomy groups, where it is proven that the map assigning to a Riemannian metric the conjugacy class of its restricted holonomy group is lower semicontinuous with respect to the $C<sup>1$-topology on the space of metrics.