Geometric phase and holonomy in the space of 2-by-2 symmetric operators
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)...
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| creator | Rondomanski, Jakub González, José D. Cojal Rabe, Jürgen P Palma, Carlos-Andres Polthier, Konrad |
| description | 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. |
| doi_str_mv | 10.48550/arxiv.2411.15038 |
| format | Article |
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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
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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
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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.</abstract><doi>10.48550/arxiv.2411.15038</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Differential Geometry Mathematics - Mathematical Physics Physics - Mathematical Physics |
| title | Geometric phase and holonomy in the space of 2-by-2 symmetric operators |
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