Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces
Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant metr...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210780 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces. Yukai Sun and Xianzhe Dai. SIGMA 16 (2020), 068, 6 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862543419385053184 |
|---|---|
| author | Sun, Yukai Dai, Xianzhe |
| author_facet | Sun, Yukai Dai, Xianzhe |
| citation_txt | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces. Yukai Sun and Xianzhe Dai. SIGMA 16 (2020), 068, 6 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact, the same result holds for a compact connected homogeneous manifold / with compact connect and semi-simple.
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| first_indexed | 2026-03-12T21:46:51Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210780 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T21:46:51Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sun, Yukai Dai, Xianzhe 2025-12-17T14:37:50Z 2020 Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces. Yukai Sun and Xianzhe Dai. SIGMA 16 (2020), 068, 6 pages 1815-0659 2020 Mathematics Subject Classification: 53C20; 53C24; 53C30 arXiv:2005.00161 https://nasplib.isofts.kiev.ua/handle/123456789/210780 https://doi.org/10.3842/SIGMA.2020.068 Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group are extremal (in fact, rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact, the same result holds for a compact connected homogeneous manifold / with compact connect and semi-simple. We are deeply grateful to Wolfgang Ziller for suggesting the more general result for the homogeneous space as well as bringing the work [15] to our attention, which considerably simplifies our previous computation as well as generalizes to the more general case of homogeneous spaces. We thank Wolfgang for many helpful discussions. Thanks are also due to Professor Yurii Nikonorov for similar remarks and useful comments. Finally, we thank the referee for the careful reading of the multiple versions of the paper and for many constructive suggestions, which have helped improve the exposition. This research is partially supported by NSFC (Y.S.) and the Simons Foundation (X.D.). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces Article published earlier |
| spellingShingle | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces Sun, Yukai Dai, Xianzhe |
| title | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_full | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_fullStr | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_full_unstemmed | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_short | Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces |
| title_sort | gromov rigidity of bi-invariant metrics on lie groups and homogeneous spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210780 |
| work_keys_str_mv | AT sunyukai gromovrigidityofbiinvariantmetricsonliegroupsandhomogeneousspaces AT daixianzhe gromovrigidityofbiinvariantmetricsonliegroupsandhomogeneousspaces |