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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.12209 |
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| _version_ | 1866917593404669952 |
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| author | Pertici, Donato Dolcetti, Alberto |
| author_facet | Pertici, Donato Dolcetti, Alberto |
| contents | We study some properties of $SU_n$ endowed with the Frobenius metric $ϕ$, which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on $SU_n$. In particular we express the distance between $P, Q \in SU_n$ in terms of eigenvalues of $P^*Q$; we compute the diameter of $(SU_n, ϕ)$ and we determine its diametral pairs; we prove that the set of all minimizing geodesic segments with endpoints $P$, $Q$ can be parametrized by means of a compact connected submanifold of $\mathfrak{su}_n$, diffeomorphic to a suitable complex Grassmannian depending on $P$ and $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some Riemannian properties of $\mathbf{SU_n}$ endowed with a bi-invariant metric Pertici, Donato Dolcetti, Alberto Differential Geometry 53C35, 15B30, 22E15 We study some properties of $SU_n$ endowed with the Frobenius metric $ϕ$, which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on $SU_n$. In particular we express the distance between $P, Q \in SU_n$ in terms of eigenvalues of $P^*Q$; we compute the diameter of $(SU_n, ϕ)$ and we determine its diametral pairs; we prove that the set of all minimizing geodesic segments with endpoints $P$, $Q$ can be parametrized by means of a compact connected submanifold of $\mathfrak{su}_n$, diffeomorphic to a suitable complex Grassmannian depending on $P$ and $Q$. |
| title | Some Riemannian properties of $\mathbf{SU_n}$ endowed with a bi-invariant metric |
| topic | Differential Geometry 53C35, 15B30, 22E15 |
| url | https://arxiv.org/abs/2402.12209 |