A Generalization of Seifert Geometry Based on the Siegel Upper Half-Space
Qing Lan
TL;DR
This work generalizes Seifert geometry by constructing a model fibering over the Siegel upper half-space ${\mathfrak{H}}_n$ via a central extension of ${\widetilde{\mathrm{Sp}}(2n,\mathbb{R})}$, yielding a homogeneous model space ${\mathcal{X}}=\widetilde{\mathrm{Sp}}(2n,\mathbb{R})/\mathrm{SU}(n)$ with a natural ${\mathbb{R}}$-fiber over ${\mathfrak{H}}_n$. A transitive action of a central extension ${\mathcal{G}}\times_{Z({\mathcal{G}})}{\mathcal{H}}{\mathcal{R}}/{\mathcal{H}}$ on ${\mathcal{X}}$ is established, with stabilizer ${\mathrm{U}}(n)/\{\pm {\rm id}\}$, and an exact sequence linking to ${\mathrm{PSp}}(2n,\mathbb{R})$ is derived. Using a Grassmannian embedding and a rank-2 normal bundle, the construction is extended from ${\mathrm{Sp}}(4,\mathbb{R})$ to general ${\mathrm{Sp}}(2n,\mathbb{R})$, and a Goetz-type product measure is employed to prove a volume formula for Seifert-like manifolds: $\mathrm{vol}(\Gamma\backslash{\mathcal{X}})=\pm\mathrm{vol}((\Gamma\cap\ker\eta)\backslash{\mathcal{R}}/{\mathcal{H}})\,\chi(\eta(\Gamma))$, in particular reducing to $\pm\chi(\eta(\Gamma))$ when the subgroup arises from ${\mathrm{PSp}}(2n,\mathbb{R})$. This generalizes Thurston’s Seifert-volume and connects representation volumes to Euler characteristics in a higher-rank symplectic setting.
Abstract
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.