  How useful it is to think about the potential energy landscape of a complex
many-body system depends in large measure on how direct the connection is to
the system's dynamics. In this paper we show that, within what we call the
potential energy landscape ensemble, it is possible to make direct connections
between the geometry of the landscape and the long-time dynamical behaviors of
systems such as supercooled liquids. We show, in particular, that the onset of
slow dynamics in such systems is governed directly by the lengths of their
geodesics - the shortest paths through their landscapes within the special
ensemble. The more convoluted and labyrinthine these geodesics are, the slower
that dynamics is. Geodesics in the landscape ensemble have sufficiently
well-defined characteristics that is straightforward to search for them
numerically, a point we illustrate by computing the geodesic lengths for an
ordinary atomic liquid and a binary glass-forming atomic mixture. We find that
the temperature dependence of the diffusion constants of these systems,
including the precipitous drop as the glass-forming system approaches its
mode-coupling transition, is predicted quantitatively by the growth of the
geodesic path lengths.
