Spectral flow and the Atiyah-Patodi-Singer index theorem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911302183550976 |
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| author | Baer, Christian Ziemke, Remo |
| author_facet | Baer, Christian Ziemke, Remo |
| contents | We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold, generalizing a result by Getzler. The spectral flow is expressed in terms of the $\hat{A}$-form of the manifold, the odd Chern character form of the family of connections, and the $ξ$-invariants of the initial and final operators. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a proof of Llarull's rigidity theorem for scalar curvature of strictly convex hypersurfaces in Euclidean space which works the same in even and odd dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04968 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral flow and the Atiyah-Patodi-Singer index theorem Baer, Christian Ziemke, Remo Differential Geometry 53C27, 58J20, 58J28, 58J30 We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold, generalizing a result by Getzler. The spectral flow is expressed in terms of the $\hat{A}$-form of the manifold, the odd Chern character form of the family of connections, and the $ξ$-invariants of the initial and final operators. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a proof of Llarull's rigidity theorem for scalar curvature of strictly convex hypersurfaces in Euclidean space which works the same in even and odd dimensions. |
| title | Spectral flow and the Atiyah-Patodi-Singer index theorem |
| topic | Differential Geometry 53C27, 58J20, 58J28, 58J30 |
| url | https://arxiv.org/abs/2512.04968 |