Hyperbolic surface ($K_{\mathrm{Gaussian}}=-1$)

A surface with a complete Riemannian metric of constant Gaussian curvature $-1$. Its local model is the Poincare half-plane model; the orientable case is a quotient by a free properly discontinuous group action of orientation-preserving isometries.

In curvature $-1$, its length is $2\mathrm{arcosh}3$. The level-two principal congruence subgroup has traces congruent to two modulo four, so the smallest hyperbolic absolute trace is six, achieved by $\left(\begin{array}{cc}5& 2\\ 2& 1\end{array}\right)$. The hyperbolic translation length formula gives the result; the shortest closed geodesic is nonsimple.

A collection of disjoint essential simple closed curves that cuts a closed genus-$g$ surface into $2g-2$ surfaces of type pair of pants. There are $3g-3$ curves. Geodesic representatives are used for Fenchel–Nielsen coordinates.

For fixed $g\ge 2$, a constant $B_g$ bounds all cuff lengths in some pants decomposition of every closed genus-$g$ hyperbolic surface. Together with Fenchel–Nielsen coordinates this implies Mumford's compactness theorem.

For a fixed pants decomposition, the positive cuff lengths and real twist parameters give coordinates on Teichmüller space. With twists measured in length units, a full Dehn twist changes ${\tau }_i$ by ${\ell }_i$; with angle units it changes the twist by $2\pi$.

A simple closed curve that is a geodesic of length $\ell$ on a hyperbolic surface has an embedded collar of half-width $w$ with $\sinh w\sinh (\ell /2)=1$. In circumference-one coordinates its metric is $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, and its conformal modulus of an annulus is $(2/\ell )\arctan (\mathrm{csch}(\ell /2))\sim \pi /\ell$.

The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.

The metric area of a hyperbolic surface. In the Poincare half-plane model the density is $dxdy/y^2$. A finite-area torsion-free quotient of genus $g$ with $r$ cusps has area $2\pi (2g-2+r)$ by the Gauss-Bonnet theorem.