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arXiv · 2510.17340

Monotonicity of holonomy groups

Linus Götzfried
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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.

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Linus Götzfried. 2025-10-20. Monotonicity of holonomy groups. https://arxiv.org/abs/2510.17340

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