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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2510.17340 |
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| _version_ | 1866918291379847168 |
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| author | Götzfried, Linus |
| author_facet | Götzfried, Linus |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity of holonomy groups Götzfried, Linus Differential Geometry 53C29 (Primary), 53C05 (Secondary) 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. |
| title | Monotonicity of holonomy groups |
| topic | Differential Geometry 53C29 (Primary), 53C05 (Secondary) |
| url | https://arxiv.org/abs/2510.17340 |