2024/11/22 by Jakub Rondomanski, Rondomanski, Jakub, José D. Cojal González +7
Mathematics · #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2411.15038
openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.