Let $n$ be a positive integer and let $k > 1$ be an integer with $k \leq n$. Consider $k$ points inside hyperbolic $n$-space $\mathbb{H}^n(R)$ for some radius $R > 0$. Prove that these $k$ points lie in fact in a totally geodesic $(k-1)$-dimensional submanifold of $\mathbb{H}^n(R)$, which is isometric to $\mathbb{H}^{k-1}(R)$.
I do not particularly know how to start this exercise. If $k = 2$, then I think the solution is pretty simple, as for $2$ points we can always take the (unique) geodesic between them if they are distinct (if they are equal, then any geodesic through the respective point will do). However, I do not know how to proceed for $k > 2$.
Since you seem to like the Lorentzian model of the hyperbolic space, here is a solution using this language (it is equivalent to the one using the Klein model suggested by Magma).
Let $q$ be a nondegenerate quadratic form on $V={\mathbb R}^{n+1}$ of signature $(n,1)$, $H$ a component of $\{v: q(v)=-1\}$ (the one contained in the future light cone). This is your model of the hyperbolic $n$-space. Take your $k$ points $x_1,...,x_{k}\in H$. Then there exists a (generically, but not always, unique) $k$-dimensional linear subspace $W\subset V$ containing these points. Then $W\cap H$ is a hyperbolic subspace of dimension $k-1$ in $H$. In the Lorentzian model, this is just a definition of a hyperbolic subspace. If you do not know this fact, it is a nice exercise in linear algebra to check that the restriction of $q$ to $W$ is again nondegenerate and has signature $(k-1,1)$.