Collinear motions in constant curvature spaces, and coplanar motions in $\mathbb{R^4}$

A collinear motion of an $n$-body problem is one in which the $n$-bodies always lie along some, possibly moving, geodesic during their evolution. We prove that every collinear motion of $n\geq 3$ bodies in a constant non-zero curvature space is necessarily contained in some fixed constant curvatur