Ideas from the theory of dynamical systems are applied to a collection of N identical point masses interacting via one-dimensional ``gravity,'' with a two-body force F=-G sgn(${x}_{a}$-${x}_{b}$). This Hamiltonian system is viewed as a geodesic flow on a curved, but conformally flat, N-dimensional space, the curvature of which exhibits certain interesting features. In the large-N limit, the Ricci tensor can be written approximately in the form ${R}_{a}^{b}$=-\ensuremath{\Lambda}${\ensuremath{\delta}}_{a}^{b}$+${B}_{a}^{b}$, where \ensuremath{\Lambda} is a smooth, negative-definite quantity which, oftentimes, can be viewed as nearly constant, and ${B}_{a}^{b}$ is a positive sum of \ensuremath{\delta} functions of codimension 1. The curvature K(u,\ensuremath{\delta}x)==${R}_{\mathrm{abcd}}$${u}^{b}$${u}^{d}$\ensuremath{\delta}${x}^{a}$\ensuremath{\delta}${x}^{c}$ associated with a small change \ensuremath{\delta}${x}^{a}$ in the N-velocity ${u}^{a}$ of the system is not negative definite. However, in the large-N limit, the smooth piece of K associated with a ``generic'' change \ensuremath{\delta}${x}^{a}$ of a generic ${u}^{a}$ is in fact negative, the fraction of ${u}^{a}$'s and \ensuremath{\delta}${x}^{a}$'s leading to a positive K decreasing at least as fast as ${N}^{\mathrm{\ensuremath{-}}1}$. If the \ensuremath{\delta}-function contributions to K could be ignored, as they can for D-dimensional ``gravity'' with D\ensuremath{\ge}2, this would imply an average ``mixing'' very much akin to the astrophysicists' ``violent relaxation'' on a time scale \ensuremath{\tau} given essentially as the free-fall time for the system. This conclusion fails, however, because of the singular contributions to K reflecting physical collisions which, for one-dimensional gravity, cannot be ignored. The similarities and differences between one- and three-dimensional ``gravitational'' systems are discussed, and it is argued that numerical simulations of one-dimensional systems need say little about real three-dimensional systems.