  Based on the study of saddle points of the potential energy landscapes of
generic classical many-particle systems, we present a necessary criterion for
the occurrence of a thermodynamic phase transition. Remarkably, this criterion
imposes conditions on microscopic properties, namely curvatures at the saddle
points of the potential, and links them to the macroscopic phenomenon of a
phase transition. We apply our result to two exactly solvable models,
corroborating that the criterion derived is not only valid, but also sharp and
useful: For both models studied, the criterion excludes the occurrence of a
phase transition for all values of the potential energy but the transition
energy. This result adds a geometrical ingredient to an established topological
condition for the occurrence of a phase transition, thereby providing an answer
to the long standing question of which topology changes in configuration space
can induce a phase transition.
