Generalized Minkowski Theorem for Tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917533481697280 |
|---|---|
| author | Liu, Hongguang Pan, Qiaoyin |
| author_facet | Liu, Hongguang Pan, Qiaoyin |
| contents | We formulate and prove a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for generalized tetrahedra in the constant-curvature Lorentzian spaces ${\rm dS}^3$ and ${\rm AdS}^3$. Four non-trivial based ${\rm SO}^+(1,2)$ holonomies, or equivalently ${\rm SL}(2,\mathbb{R})$ spin lifts, determine intrinsic face normals, a dihedral Gram matrix $G$, and oriented triple products of intrinsic face normals. Under closure, nondegeneracy, and the outward convex branch condition, these data reconstruct a unique strictly convex tetrahedron up to ambient isometry. The sign of $\det G$ selects the de Sitter or anti-de Sitter model, and the prescribed holonomies are exactly the based Levi-Civita face holonomies of the reconstructed tetrahedron. The extrinsic face normals also define a polar-dual projective tetrahedron. In particular, the all-null AdS sector gives ideal dual tetrahedra, and the all-timelike AdS sector gives hyperideal dual tetrahedra. In the all-spacelike sector, changing to ${\rm SU}(2)$ real form recovers the reconstruction theorem for Euclidean spherical and hyperbolic tetrahedra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26410 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Minkowski Theorem for Tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$ Liu, Hongguang Pan, Qiaoyin Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry We formulate and prove a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for generalized tetrahedra in the constant-curvature Lorentzian spaces ${\rm dS}^3$ and ${\rm AdS}^3$. Four non-trivial based ${\rm SO}^+(1,2)$ holonomies, or equivalently ${\rm SL}(2,\mathbb{R})$ spin lifts, determine intrinsic face normals, a dihedral Gram matrix $G$, and oriented triple products of intrinsic face normals. Under closure, nondegeneracy, and the outward convex branch condition, these data reconstruct a unique strictly convex tetrahedron up to ambient isometry. The sign of $\det G$ selects the de Sitter or anti-de Sitter model, and the prescribed holonomies are exactly the based Levi-Civita face holonomies of the reconstructed tetrahedron. The extrinsic face normals also define a polar-dual projective tetrahedron. In particular, the all-null AdS sector gives ideal dual tetrahedra, and the all-timelike AdS sector gives hyperideal dual tetrahedra. In the all-spacelike sector, changing to ${\rm SU}(2)$ real form recovers the reconstruction theorem for Euclidean spherical and hyperbolic tetrahedra. |
| title | Generalized Minkowski Theorem for Tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$ |
| topic | Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry |
| url | https://arxiv.org/abs/2605.26410 |