= Solution
For the full unit-radius <two-dimensional de Sitter spacetime>, the coordinate ranges are
$$
\boxed{t\in\mathbb R,\qquad \chi\in\mathbb R/(2\pi\mathbb Z).}
$$
One can use $0\leq\chi<2\pi$ with endpoints identified. If the identification is omitted, the same line element describes the universal covering spacetime instead. The periodicity follows from the <hyperboloid> embedding
$$
X^0=\sinh t,\qquad X^1=\cosh t\cos\chi,\qquad X^2=\cosh t\sin\chi,
\qquad (X^0)^2-(X^1)^2-(X^2)^2=-1.
$$
The induced ambient <Minkowski metric> is precisely the given <Lorentzian metric>.
Using an <affine parameter> $s$, the <geodesic Lagrangian> is $\tfrac12(\dot t^2-\cosh^2t\,\dot\chi^2)$. The nonzero <Christoffel symbols> are
$$
\Gamma^t{}_{\chi\chi}=\sinh t\cosh t,
\qquad \Gamma^\chi{}_{t\chi}=\Gamma^\chi{}_{\chi t}=\tanh t.
$$
Thus the <geodesic equations> and their <first integrals> are
$$
\boxed{\ddot t+\sinh t\cosh t\,\dot\chi^2=0,
\qquad \ddot\chi+2\tanh t\,\dot t\dot\chi=0,}
$$
$$
J=\cosh^2t\,\dot\chi=\mathrm{constant},
\qquad \dot t^2-\frac{J^2}{\cosh^2t}=\varepsilon,
\qquad \varepsilon\in\{1,0,-1\}.
$$
Here $\varepsilon=1$ for unit-speed <timelike geodesics>, $0$ for <null geodesics> and $-1$ for unit-speed <spacelike geodesics>. The conserved $J$ is the <geodesic conserved quantity from a Killing vector> for the angular symmetry.
All <geodesics of two-dimensional de Sitter spacetime> can be written especially simply in the embedding. If $P$ is the initial point and $V$ the initial tangent, with $P^2=-1$, $P\cdot V=0$, $V^2=\varepsilon$, the geodesic has normal ambient acceleration $X''=\varepsilon X$. Therefore
$$
X(s)=
\begin{cases}
P\cosh s+V\sinh s,&\varepsilon=1,\\
P+sV,&\varepsilon=0,\\
P\cos s+V\sin s,&\varepsilon=-1.
\end{cases}
$$
These curves stay in the two-plane spanned by $P$ and $V$, preserve both constraints and have no tangential acceleration. They consequently solve the <geodesic equation> and exhaust all initial directions.
An <isometry> can move the chosen point to $P=(0,1,0)$, namely $(t,\chi)=(0,0)$. For future-directed <timelike geodesics> take $V=(\cosh\alpha,0,\sinh\alpha)$, obtaining
$$
\sinh t=\cosh\alpha\sinh s,
\qquad \tan\chi=\sinh\alpha\tanh s.
$$
The continuous angular branch has $|\chi|<\pi/2$, and every real $\alpha$ gives a distinct timelike direction. The two <null geodesics> have $V=(1,0,\pm1)$ and
$$
\sinh t=s,\qquad \tan\chi=\pm s.
$$
For <spacelike geodesics> take $V=(\sinh\alpha,0,\cosh\alpha)$, giving
$$
\sinh t=\sinh\alpha\sin s,
\quad \cosh t\cos\chi=\cos s,
\quad \cosh t\sin\chi=\cosh\alpha\sin s.
$$
They are closed with period $2\pi$ in affine arclength and $|t|\leq|\alpha|$. In particular, $\alpha=0$ is the equatorial geodesic $t=0$. These three families, with reversals of parametrization allowed, are the requested complete set through the chosen point.
The <conformal time> substitution gives
$$
T=2\arctan(e^t)-\frac\pi2,
\qquad \sin T=\tanh t,
\qquad \cos T=\operatorname{sech}t,
\qquad \frac{dT}{dt}=\operatorname{sech}t.
$$
Hence
$$
\boxed{ds^2=\sec^2T(dT^2-d\chi^2),
\qquad -\frac\pi2<T<\frac\pi2,\quad\chi\sim\chi+2\pi.}
$$
Multiplying by $\cos^2T$ gives the regular <conformal cylinder of two-dimensional de Sitter spacetime>. Adding $T=\pm\pi/2$ completes the <conformal compactification>; both boundary circles are spacelike. <Null geodesics> are straight lines $\chi=\pm T$ through the origin. The other families have the useful diagram equations
$$
\chi(T)=\arcsin(\tanh\alpha\sin T)\quad\text{for timelike geodesics},
\qquad T(\chi)=\arcsin(\tanh\alpha\sin\chi)\quad\text{for spacelike geodesics}.
$$
The image sketches representative directions from each family; the formulas allow all intermediate directions.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-63-de-sitter-geodesics.png]
{title=Geodesics through one point and observer horizons on the two-dimensional de Sitter conformal cylinder}
For the complete observer $\chi=0$, let $d(\chi,0)\in[0,\pi]$ be the shortest angular distance on the circle. Since light travels one unit of angular distance per unit <conformal time>, an event at $(T,\chi)$ can signal to the observer at some finite future time precisely when
$$
\boxed{d(\chi,0)<\frac\pi2-T.}
$$
Equality defines its future <observer event horizon>. Likewise an event can receive a signal emitted by the observer precisely when
$$
\boxed{d(\chi,0)<T+\frac\pi2,}
$$
and equality defines the past horizon. In the lift $-\pi\leq\chi\leq\pi$, their branches are $\chi=\pm(\pi/2-T)$ and $\chi=\pm(T+\pi/2)$. The common communication region is the diamond $|\chi|+|T|<\pi/2$, the observer's <static patch of de Sitter spacetime>. These are the <observer horizons in two-dimensional de Sitter spacetime>. Every complete timelike observer has analogous horizons after a <de Sitter isometry>; they depend on the observer, rather than enclosing an intrinsic black-hole region.
Despite these horizons, each constant-$T$ circle is a <Cauchy hypersurface> of the full spacetime: any inextendible <causal curve> crosses it exactly once. The global spacetime is therefore <globally hyperbolic> and has no <Cauchy horizons>. The conformal boundaries lie at infinite proper time for complete <timelike geodesics> and infinite affine distance for complete <null geodesics>, so their finite height in the diagram is not a physical incompleteness.
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