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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.05245 |
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| _version_ | 1866915096048959488 |
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| author | Lan, Qing |
| author_facet | Lan, Qing |
| contents | Parallel to $\widetilde{\mathrm{SL}(2,\mathbb{R})}$-geometry fibering over the hyperbolic plane, we construct a geometry fibering over the Siegel upper half-space $\mathrm{Sp}(2n,\mathbb{R})\curvearrowright {\mathfrak{H}}_n$, and provide a volume formula for some manifolds with this geometry. For $n=2$, a prototype is constructed via the normal bundle of an equivariant embedding into a Grassmannian manifold. It turns out that this geometry is the homogeneous space given by a central extension of $\widetilde{\mathrm{Sp}(2n,\mathbb{R})}$, modulo its maximal compact subgroup. After fixing a convention for the invariant measure, the volume of a "Seifert-like" closed manifold of this geometry is given by the length of the fiber circle times the Euler characteristic of the base manifold, up to a sign. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Generalization of Seifert Geometry Based on the Siegel Upper Half-Space Lan, Qing Geometric Topology Differential Geometry Parallel to $\widetilde{\mathrm{SL}(2,\mathbb{R})}$-geometry fibering over the hyperbolic plane, we construct a geometry fibering over the Siegel upper half-space $\mathrm{Sp}(2n,\mathbb{R})\curvearrowright {\mathfrak{H}}_n$, and provide a volume formula for some manifolds with this geometry. For $n=2$, a prototype is constructed via the normal bundle of an equivariant embedding into a Grassmannian manifold. It turns out that this geometry is the homogeneous space given by a central extension of $\widetilde{\mathrm{Sp}(2n,\mathbb{R})}$, modulo its maximal compact subgroup. After fixing a convention for the invariant measure, the volume of a "Seifert-like" closed manifold of this geometry is given by the length of the fiber circle times the Euler characteristic of the base manifold, up to a sign. |
| title | A Generalization of Seifert Geometry Based on the Siegel Upper Half-Space |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2501.05245 |