nLab hyperbolic 3-manifold

Contents

Context

Riemannian geometry

Riemannian geometry

Contents

Idea

A Riemannian manifold which is both a 3-manifold and a hyperbolic manifold is a hyperbolic 3-manifold.

Equivalently this is a Riemannian manifold which is isometric to the quotient space $\mathbb{H}^3/ \Gamma$ of hyperbolic 3-space by the action of a torsion-free discrete group $\Gamma$ acting by isometries.

Properties

Volume conjecture

For the moment see at volume conjecture.