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 curvature space of dimension at most three. For natural systems admitting energy and angular momentum integrals, we further show that every such motion is necessarily a relative equilibrium: evolving under uniform rotation while maintaining constant mutual distances. These results extend certain classical results on collinear motions in Euclidean spaces, initiated by P. Pizzetti, to general spaces of constant curvature. The same methods also yield a partial answer to a question of A. Wintner. Namely, we show that any concave and always coplanar motion in the Euclidean 4-body problem, where the moving plane containing the bodies sweeps out an $\mathbb{R}^4$, must be self-similar: maintaining constant mutual distance ratios over the motion.