Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index
Every closed connected Riemannian spin manifold of non-zero Â-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. Thi...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/214094 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index. Thomas Tony. SIGMA 21 (2025), 072, 7 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862709904700080128 |
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| author | Tony, Thomas |
| author_facet | Tony, Thomas |
| citation_txt | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index. Thomas Tony. SIGMA 21 (2025), 072, 7 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Every closed connected Riemannian spin manifold of non-zero Â-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. This provides a criterion for the existence of a parallel spinor on a finite covering and yields that every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy.
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| first_indexed | 2026-03-19T10:47:53Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214094 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T10:47:53Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tony, Thomas 2026-02-19T11:12:58Z 2025 Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index. Thomas Tony. SIGMA 21 (2025), 072, 7 pages 1815-0659 2020 Mathematics Subject Classification: 53C29; 58J20; 53C21; 58B34 arXiv:2411.03882 https://nasplib.isofts.kiev.ua/handle/123456789/214094 https://doi.org/10.3842/SIGMA.2025.072 Every closed connected Riemannian spin manifold of non-zero Â-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. This provides a criterion for the existence of a parallel spinor on a finite covering and yields that every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy. I thank Bernd Ammann for coming up with this interesting question and for his hospitality during my short-term visit to Regensburg at the SFB 1085 Higher Invariants. I also thank my advisor, Rudolf Zeidler, for his continuous support, as well as Thorsten Hertl and the anonymous referees for their valuable comments. Funded by the European Union (ERC Starting Grant 101116001– COMSCAL)1 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– Project-ID 427320536– SFB1442, as well as under Germany’s Excellence Strategy EXC 2044 390685587, Mathematics M¨unster: Dynamics–Geometry–Structure. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index Article published earlier |
| spellingShingle | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index Tony, Thomas |
| title | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index |
| title_full | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index |
| title_fullStr | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index |
| title_full_unstemmed | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index |
| title_short | Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index |
| title_sort | ricci-flat manifolds, parallel spinors and the rosenberg index |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214094 |
| work_keys_str_mv | AT tonythomas ricciflatmanifoldsparallelspinorsandtherosenbergindex |