Hyperbolic space is complete

The point $(0,1)$ lies in $H$. If $p=(x,y)\in H$ and $\parallel q-p{\parallel }_2<y\mathrm{/}2$, then [F1] gives $\mathrm{\mid }{q}_y-y\mathrm{\mid }\le \parallel q-p{\parallel }_2<y\mathrm{/}2$, whence $q_y>y\mathrm{/}2>0$ and $q\in H$; hence $H$ is open. For the coefficient $f(x,y)=y^{-2}$, define $c_0=1$ and $c_{r+1}=-(r+2)c_r$. Induction with the product and quotient rules in [F1] gives ${\mathrm{\partial }}_y^{r}f=c_r{y}^{-r-2}$ for every $r\ge 0$; any iterated partial containing ${\mathrm{\partial }}_x$ is zero. The one-variable function $h_r(y)=c_r{y}^{-r-2}$ is differentiable and hence continuous on $(0,\mathrm{\infty })$. Since $\mathrm{\mid }{q}_y-y\mathrm{\mid }\le \parallel q-(x,y){\parallel }_2$ by [F1], continuity of $h_r$ makes $(x,y)\mapsto h_r(y)$ continuous on $H$. Thus the $C^k$ criterion in [F1] for every $k$ makes $f$ smooth. Finally, for $v=(v_1,v_2)\mathrm{\ne }0$, $g_{(x,y)}(v,v)=\frac{v_1^2+v_2^2}{y^2}>0.$ Thus [F1] makes $(H,g)$ a nonempty boundaryless Riemannian two-manifold.

For $p,q\in H$ and $s\in [0,1]$, the second coordinate of $(1-s)p+sq$ is the positive number $(1-s)p_y+s{q}_y$. The coordinate polynomials in $s$ are continuous and their pair has image in $H$, so this segment is a path from $p$ to $q$. Thus $H$ is path-connected, and [F2] makes it connected.

The curves in [F3] are defined for every $t\in \mathbb{R}$. Their hyperbolic speeds can also be read directly. The vertical curve has $g(\dot{\gamma },\dot{\gamma })=\frac{k^2{b}^2{e}^{2kt}}{b^2{e}^{2kt}}=k^2\mathrm{.}$ For the semicircle put $U=kt+c$, $T=\tanh U$, and $S=\mathrm{sech}U$. Then $\dot{x}=Rk{S}^2,\dot{y}=-RkST,y=RS,$ and $S^2+T^2=1$, so $g(\dot{\gamma },\dot{\gamma })=\frac{R^2{k}^2{S}^4+R^2{k}^2{S}^2{T}^2}{R^2{S}^2}=k^2\mathrm{.}$ In particular $\mathrm{\mid }k\mathrm{\mid }=1$ gives hyperbolic arclength parametrizations on all of $\mathbb{R}$; arbitrary $k\mathrm{\ne }0$ gives complete affine constant-speed parametrizations.

Fix initial data and let $\gamma :I\to H$ be its unique maximal geodesic. If its initial velocity is zero, the constant geodesic with that initial data is defined on $\mathbb{R}$, so maximality gives $I=\mathbb{R}$. Otherwise [F3] identifies $\gamma$ on $I$ with a restriction of one of the two curves in step 1.3, and [F3] proves that the corresponding curve on all of $\mathbb{R}$ is a geodesic. It is therefore an extension of $\gamma$ unless $I=\mathbb{R}$. Maximality forces $I=\mathbb{R}$ in every case, and [F4] makes $(H,g)$ geodesically complete.

Steps 1.1 and 1.2 verify the nonempty, connected, boundaryless Riemannian hypotheses of [F5], and step 2.1 verifies its geodesic-completeness condition. The implication from that condition to metric completeness in [F5] therefore makes $(H,d_g)$ complete. The explicit classification and extension calculation in steps 1.1--2.1 use no choice; ${\mathrm{A}\mathrm{C}}_{\omega }$ is used only through the current maximal-geodesic completeness convention [F4] and Hopf--Rinow [F5].