= Solution
Take both curvature radii to be one. Coordinate ranges depend on the global identification, which a line element alone cannot fix. For the ordinary <two-dimensional de Sitter spacetime>,
$$
\boxed{t\in\mathbb R,\qquad\chi\in\mathbb R/(2\pi\mathbb Z).}
$$
This is the unit hyperboloid $X_0^2-X_1^2-X_2^2=-1$ in ambient signature $(+--)$, parametrized by $X_0=\sinh t$, $X_1=\cosh t\cos\chi$, $X_2=\cosh t\sin\chi$. Unwrapping $\chi$ instead gives its <universal cover>.
For <two-dimensional anti-de Sitter spacetime>, the embedding $X_0^2+X_1^2-X_2^2=1$ in ambient signature $(++-)$ is parametrized by $X_0=\cosh r\cos t$, $X_1=\cosh r\sin t$, $X_2=\sinh r$. On this hyperboloid $r\in\mathbb R$ and $t$ is periodic modulo $2\pi$, creating closed timelike circles. The physically usual <universal cover> removes that periodicity:
$$
\boxed{r\in\mathbb R,\qquad t\in\mathbb R\quad\text{on the covering AdS spacetime}.}
$$
Both signs of $r$ are needed for the full two-dimensional spatial line. Taking $r\geq0$ without an additional construction would omit one of its two ends. The following causal comparison uses the AdS <universal cover>.
Both models are maximally symmetric <Lorentzian manifolds>, with <constant sectional curvature> and no curvature singularities. In the paper's convention their <Ricci scalars> are respectively $+2$ and $-2$. The de Sitter spatial scale factor $a(t)=\cosh t$ contracts to a nonzero minimum and then expands; $t=0$ is not a big-bang singularity. The AdS metric is static, and $\partial_t$ is timelike everywhere because $g_{tt}=\cosh^2r>0$.
The embeddings give a concise classification of all affinely parametrized <geodesics>. <Geodesic> <acceleration> in the ambient space is normal to the hyperboloid, hence proportional to $X$. Differentiating the constraint and setting $X'^2=\kappa$, with $\kappa=1,0,-1$ for timelike, null and spacelike curves, gives
$$
X''=\kappa X\quad\text{in de Sitter},\qquad
X''=-\kappa X\quad\text{in anti-de Sitter}.
$$
Every solution stays in the two-plane spanned by its initial point $P$ and tangent $V$, so every <geodesic> is a plane section through the ambient origin. For de Sitter, these are $P\cosh s+V\sinh s$ for timelike curves, $P+sV$ for null curves, and $P\cos s+V\sin s$ for spacelike curves. Thus <timelike geodesics> and <null geodesics> are complete and nonclosed, while <spacelike geodesics> are closed circles on the ordinary hyperboloid. These are the <geodesics of two-dimensional de Sitter spacetime>.
For AdS the roles of the circular and hyperbolic solutions are reversed: timelike curves are $P\cos s+V\sin s$, null curves are $P+sV$, and spacelike curves are $P\cosh s+V\sinh s$. Timelike curves close after <proper time> $2\pi$ on the hyperboloid, but their lifts on the <universal cover> continue to ever later time rather than forming closed curves. All three kinds extend for arbitrary <affine parameter>. This is <geodesics of two-dimensional anti-de Sitter spacetime>, not an assertion that <geodesics> reach the <conformal boundary> in finite physical affine length.
The coordinate first integrals make the behavior concrete. In de Sitter, spatial rotational symmetry gives a conserved $P_\chi=\cosh^2t\,d\chi/ds$, and normalization gives
$$
\left(\frac{dt}{ds}\right)^2=\kappa+\frac{P_\chi^2}{\cosh^2t},\qquad
\frac{d\chi}{ds}=\frac{P_\chi}{\cosh^2t}.
$$
Constant-$\chi$ observers are <timelike geodesics>. Nontrivial <null geodesics> have <affine parameter> proportional to $\sinh t$, which diverges at both $t\to\pm\infty$. In AdS the conserved static <energy> is $E=\cosh^2r\,dt/ds$, with
$$
\left(\frac{dr}{ds}\right)^2=\frac{E^2}{\cosh^2r}-\kappa.
$$
For timelike curves, $E\geq1$ and
$$
\sinh r=\sqrt{E^2-1}\sin(s-s_0).
$$
They oscillate through the center and never reach spatial infinity; $r=0$ is the $E=1$ <geodesic>. For null curves, $\lambda=\pm\sinh r/E+\text{constant}$, so both ends are at infinite <affine parameter>. Spacelike AdS <geodesics> likewise extend to the two spatial ends.
For the <conformal structure>, introduce de Sitter <conformal time>
$$
\eta=\arctan(\sinh t),\qquad -\frac\pi2<\eta<\frac\pi2,\qquad
\boxed{ds^2=\sec^2\eta\,(d\eta^2-d\chi^2).}
$$
Multiplying by $\cos^2\eta$ gives the <conformal cylinder of two-dimensional de Sitter spacetime>. The past and future conformal infinities are spacelike circles at $\eta=\mp\pi/2$. Null curves are $\chi=\pm\eta+\text{constant}$ modulo $2\pi$. Although the conformal-time interval is finite, their physical <affine parameters> and timelike observers' proper times are infinite at its ends. Constant-$\eta$ circles are <Cauchy hypersurfaces>; global de Sitter is <globally hyperbolic>.
For AdS put
$$
\psi=\arctan(\sinh r),\qquad -\frac\pi2<\psi<\frac\pi2,\qquad
\boxed{ds^2=\sec^2\psi\,(dt^2-d\psi^2).}
$$
The <conformal strip of two-dimensional anti-de Sitter spacetime> has unbounded time and timelike <conformal boundaries> at $\psi=\pm\pi/2$. Null curves have $t\pm\psi=\text{constant}$. A signal from any finite $r$ reaches the center in coordinate time $|\psi|<\pi/2$, and a null curve crosses the entire conformal strip in time $\pi$. This finite coordinate travel time coexists with infinite physical affine length to the boundary. The covering spacetime has no <closed timelike curves>, but is not <globally hyperbolic>: causal curves can arrive from timelike infinity without meeting a proposed initial slice. Field evolution consequently requires boundary conditions as well as initial data. Unwrapping time removes <closed timelike curves>, not the timelike boundary.
The contrast in <observer event horizons> follows directly from these null curves. For the complete de Sitter observer $\chi=0$, let $d(\chi,0)\in[0,\pi]$ be shortest angular distance. An event can send a signal to this observer before its future endpoint precisely when
$$
d(\chi,0)<\frac\pi2-\eta.
$$
Equality is its future <observer event horizon>. It can receive a signal emitted by the observer after its past endpoint precisely when $d(\chi,0)<\eta+\pi/2$, whose equality is the past horizon. In the observer's fundamental domain these horizons are null lines
$$
\chi=\pm(\pi/2-\eta),\qquad\chi=\pm(\eta+\pi/2).
$$
Their intersection encloses the observer's <static patch of de Sitter spacetime>, the diamond $d(\chi,0)<\pi/2-|\eta|$. They are observer-dependent <cosmological horizons>, not curvature singularities or <Cauchy horizons>. Every inertial observer has the corresponding horizons by de Sitter symmetry.
For the complete AdS static observer at $r=0$, every event at finite $r$ can send a signal to the observer, and can receive one, because time is unbounded and the coordinate distance $|\psi|$ is finite. Thus there is no corresponding global static-observer event horizon. The lapse never vanishes in these coordinates. Restricted accelerated-observer patches can have observer horizons, but those are distinct from the global static model. On the original periodic-time AdS hyperboloid, the more serious issue is <closed timelike curves>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-61-conformal-models.png]
{title=Conformal cylinder of de Sitter and covering anti-de Sitter strip, showing null geodesics, observer horizons and causal boundaries}
The figure identifies the de Sitter spatial edges and displays only a finite time window of the AdS covering strip, whose time continues indefinitely. Solid red curves are <null geodesics>; the oscillating AdS curve is a <timelike geodesic>. The shaded de Sitter diamond is the static observer's two-way communication region, not the entire spacetime.
The concise comparison is \b[de Sitter has spacelike conformal infinities and cosmological observer horizons; covering AdS has timelike <conformal boundaries> and no horizon for an eternal global static observer]. Both are constant-curvature and geodesically complete, but their <geodesic> recurrence, global topology, causal boundaries and initial-value properties differ.
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