Geometric phase and holonomy in the space of 2-by-2 symmetric operators
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929600982941696 |
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| author | Rondomanski, Jakub González, José D. Cojal Rabe, Jürgen P. Palma, Carlos-Andres Polthier, Konrad |
| author_facet | Rondomanski, Jakub González, José D. Cojal Rabe, Jürgen P. Palma, Carlos-Andres Polthier, Konrad |
| contents | We present a non-trivial metric tensor field on the space of 2-by-2 real-valued, symmetric matrices whose Levi-Civita connection renders frames of eigenvectors parallel. This results in fundamental reimagining of the space of symmetric matrices as a curved manifold (rather than a flat vector space) and reduces the computation of eigenvectors of one-parameter-families of matrices to a single computation of eigenvectors at an initial point, while the rest are obtained by the parallel transport ODE. Our work has important applications to vibrations of physical systems whose topology is directly explained by the non-trivial holonomy of the spaces of symmetric matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric phase and holonomy in the space of 2-by-2 symmetric operators Rondomanski, Jakub González, José D. Cojal Rabe, Jürgen P. Palma, Carlos-Andres Polthier, Konrad Differential Geometry Mathematical Physics We present a non-trivial metric tensor field on the space of 2-by-2 real-valued, symmetric matrices whose Levi-Civita connection renders frames of eigenvectors parallel. This results in fundamental reimagining of the space of symmetric matrices as a curved manifold (rather than a flat vector space) and reduces the computation of eigenvectors of one-parameter-families of matrices to a single computation of eigenvectors at an initial point, while the rest are obtained by the parallel transport ODE. Our work has important applications to vibrations of physical systems whose topology is directly explained by the non-trivial holonomy of the spaces of symmetric matrices. |
| title | Geometric phase and holonomy in the space of 2-by-2 symmetric operators |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2411.15038 |