  Hard-sphere models exhibit many of the same kinds of supercooled-liquid
behavior as more realistic models of liquids, but the highly non-analytic
character of their potentials makes it a challenge to think of that behavior in
potential-energy-landscape terms. We show here that it is possible to calculate
an important topological property of hard-sphere landscapes, the geodesic
pathways through those landscapes, and to do so without artificially
coarse-graining or softening the potential. We show, moreover, that the rapid
growth of the lengths of those pathways with increasing packing fraction
quantitatively predicts the precipitous decline in diffusion constants in a
glass-forming hard-sphere mixture model. The geodesic paths themselves can be
considered as defining the intrinsic dynamics of hard spheres, so it is also
revealing to find that they (and therefore the features of the underlying
potential-energy landscape) correctly predict the occurrence of dynamic
heterogeneity and non-zero values of the non-Gaussian parameter. The success of
these landscape predictions for the dynamics of such a singular model
emphasizes that there is more to potential energy landscapes than is revealed
by looking at the minima and saddle points.
