Let $\mathbb{H}^n$ be the hyperbolic space defined as warped product: $$ g_{\mathbb{H}^n} = dr^2 + \sinh(r)^2 g_{\mathbb{S}^{n-1}}. $$ What is the easiest way to show that there exist at least one line in $\mathbb{H}$?
A line $\gamma$ is a unit speed geodesic $\gamma : \mathbb{R} \to \mathbb{H}^n$ such that $$ dist \big(\gamma(t_1), \gamma(t_2)\big) = |t_1 - t_2| \qquad \text{for every } t_1, t_2 \in \mathbb{R}. $$
The set of fixed point of an isometry is totally geodesic. Now, if $(S,N)$ are two antipodal points in the unit sphere $S$ there exist an isometry $\sigma $ of $S$ wich fixes exactly these two points (the restriction of a reflexion to $S$). by the definition of the metric the map $T(r,u)=(r, \sigma(u))$ is an isometry whose fixed point set is $\bf H^1$, i.e. a line. The same argument proves that $\bf H^k$ is totally geodesic in $\bf H^n$ .