  We study the global geometry of the energy landscape of a simple model of a
self-gravitating system, the self-gravitating ring (SGR). This is done by
endowing the configuration space with a metric such that the dynamical
trajectories are identified with geodesics. The average curvature and curvature
fluctuations of the energy landscape are computed by means of Monte Carlo
simulations and, when possible, of a mean-field method, showing that these
global geometric quantities provide a clear geometric characterization of the
collapse phase transition occurring in the SGR as the transition from a flat
landscape at high energies to a landscape with mainly positive but fluctuating
curvature in the collapsed phase. Moreover, curvature fluctuations show a
maximum in correspondence with the energy of a possible further transition,
occurring at lower energies than the collapse one, whose existence had been
previously conjectured on the basis of a local analysis of the energy landscape
and whose effect on the usual thermodynamic quantities, if any, is extremely
weak. We also estimate the largest Lyapunov exponent $\lambda$ of the SGR using
the geometric observables. The geometric estimate always gives the correct
order of magnitude of $\lambda$ and is also quantitatively correct at small
energy densities and, in the limit $N\to\infty$, in the whole homogeneous
phase.
