Monotonicity of holonomy groups

TL;DR

This paper establishes a monotonicity property of holonomy groups under metric convergence, showing that limits preserve certain holonomy restrictions, which implies lower semicontinuity of the holonomy map.

Contribution

It proves that holonomy groups are monotonic under $C^0$ convergence of connections and $C^1$ convergence of metrics, extending understanding of holonomy stability.

Findings

Holonomy groups are contained in a closed group under $C^0$ limits of connections.

Limit metrics with special holonomy retain their restricted holonomy properties.

The holonomy assignment map is lower semicontinuous with respect to inclusion.

Abstract

We prove the following monotonicity result for the holonomy group: Given a sequence of metric connections converging in $C^0$ such that all its members have holonomy contained in a closed group $H$, also their limit connection needs to have holonomy contained in $H$. As a corollary, for a sequence of Riemannian metrics converging in $C^1$ and having special restricted holonomy, their limit metric must also have special restricted holonomy. In particular, this implies that the map assigning to Riemannian metrics on a manifold the conjugacy classes of their restricted holonomy groups is lower semicontinuous with respect to the order relation given by inclusion of representatives.