A generalization of Seifert geometry based on the Siegel upper half-space

The Seifert geometry, $\widetilde{\mathrm{S}\mathrm{L}(2,\mathbb{R})}$-geometry, is one of Thurston’s eight $3$-dimensional geometries. It fibers over the hyperbolic plane ${\mathbb{H}}^2$, which is a special case of the Siegel upper half-space $\mathrm{S}\mathrm{p}(2n,\mathbb{R})\curvearrowright {\mathfrak{H}}_n$. In this paper we construct an analogous geometry fibering over the Siegel upper half-space, 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 $\widetilde{\mathrm{S}\mathrm{p}(2n,\mathbb{R})}$, modulo its maximal compact subgroup. The volume of a Siegel–Seifert closed manifold of this geometry is shown to be the length of the fiber circle times the Euler characteristic of the base manifold, up to a sign. Examples of Siegel–Seifert manifolds are provided, and it is shown that the volume of representations for this geometry is constant on every path-connected component of the representation space.