Hyperbolic three-space

Definition

Hyperbolic three-space $\mathbb H^3$ is the simply connected complete $3$ -dimensional Riemannian manifold of constant sectional curvature $-1$ , unique up to isometry.

Standard models

In the upper-half-space model,

\mathbb H^3=\{(z,r):z\in\mathbb C,\ r>0\}, \qquad ds^2=\frac{|dz|^2+dr^2}{r^2}.

In the hyperboloid model, using the $(-+++)$ form fixed for Minkowski space,

\mathbb H^3=\{v\in\mathbb R^{1,3}:q(v)=-1,\ t>0\},

with Riemannian metric $\eta$ restricted to tangent spaces.

A third model is the space of positive-definite Hermitian $2\times2$ matrices $H$ with $\det H=1$ . The identification with the hyperboloid uses the Hermitian matrix model.

Boundary and symmetry

The ideal or conformal boundary is

\partial_\infty\mathbb H^3\cong S^2\cong\mathbb{CP}^1.

The orientation-preserving isometry group is $PSL(2,\mathbb C)$ , and its boundary action is the Möbius action. Discrete subgroups of this group are Kleinian groups; torsion-free discrete subgroups give hyperbolic $3$ -manifolds as quotients.

References

John G. Ratcliffe, Foundations of Hyperbolic Manifolds, 3rd ed., Springer, 2019, Chapters 3–4. Publisher record.
Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983, Chapter 7. Publisher record.