Prescribed Riemannian Symmetries
Given a smooth free action of a compact connected Lie group 𝐺 on a smooth compact manifold 𝑀, we show that the space of 𝐺-invariant Riemannian metrics on 𝑀 whose automorphism group is precisely 𝐺 is open and dense in the space of all 𝐺-invariant metrics, provided the dimension of 𝑀 is ''su...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211319 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Prescribed Riemannian Symmetries. Alexandru Chirvasitu. SIGMA 17 (2021), 030, 17 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | Given a smooth free action of a compact connected Lie group 𝐺 on a smooth compact manifold 𝑀, we show that the space of 𝐺-invariant Riemannian metrics on 𝑀 whose automorphism group is precisely 𝐺 is open and dense in the space of all 𝐺-invariant metrics, provided the dimension of 𝑀 is ''sufficiently large'' compared to that of 𝐺. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way, we also show, under less restrictive conditions on both dimensions and actions, that the space of 𝐺-invariant metrics whose automorphism groups preserve the 𝐺-orbits is dense 𝐺δ in the space of all 𝐺-invariant metrics.
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| ISSN: | 1815-0659 |