In Thurston's notes on hyperbolic geometry, why does a geodesic always belong to a 2D subspace?

Question

On page 12 of [1] (in the subsection titled Trigonometry), Thurston derives the equation of a geodesic in the hyperboloid model of hyperbolic geometry. The hyperboloid in the hyperboloid model is given by $x^2 + y^2 - t^2 = 1$. The metric tensor is given by $g_{ij}=\begin{cases}1, & i = j < 3 \\ -1, & i = j = 3 \\ 0, & \text{otherwise} \end{cases}$. At some point, he states that "any geodesic must lie in a two-dimensional subspace", and from that claim he derives that $\ddot{X}_t = -(s/r)^2X_t$ (where we can set $r = s = 1$ without any loss of generality, because they are constants). Thurston appears to be using a general fact about geodesics in Riemannian manifolds to conclude that a geodesic in the hyperboloid model of hyperbolic geometry is always the intersection of the hyperboloid with a plane that passes through the origin. What general fact is he using? By what fact in Riemannian geometry must a geodesic lie on a 2-dimensional vector subspace?

[1] - http://library.msri.org/books/gt3m/PDF/2.pdf