Finding a metric of constant negative curvature on cylinder over a torus ($\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$) I read that a counterexample to show that compacity as a hypothesis in Preissman's theorem is a necessary condition is the manifold  $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$, which admits a complete metric of constant negative sectional curvature. I realize for this to be true, we must be able to see it as a quotient of $\mathbb{H}^3$, but I don't know why that's true. Can anyone shed some light on this example? What metric is this, explicity?

Consider the upper half-space model of ${\mathbb H}^3$ whose elements are identified with pairs $(z,t), z\in {\mathbb C}, t>0$. Take the group $\Gamma$ of isometries of ${\mathbb H}^3$ generated by the Euclidean translations
$$
a: z\mapsto z+1, b: z\mapsto z+i.
$$
This group is isomorphic to ${\mathbb Z}^2$, it acts on ${\mathbb H}^3$
properly discontinuously, preserving its (topological) product decomposition ${\mathbb C}\times (0,\infty)$. Since ${\mathbb C}/\Gamma\cong T^2$, we obtain a diffeomorphism
$$
M={\mathbb H}^3/\Gamma\to T^2\times (0,\infty). 
$$
Thus, $M$ is your hyperbolic manifold.
