  The Hamiltonian dynamics associated to classical, planar, Heisenberg XY
models is investigated for two- and three-dimensional lattices. Besides the
conventional signatures of phase transitions, here obtained through time
averages of thermodynamical observables in place of ensemble averages,
qualitatively new information is derived from the temperature dependence of
Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests
to consider other observables of geometric meaning tightly related with the
largest Lyapunov exponent. The numerical computation of these observables -
unusual in the study of phase transitions - sheds a new light on the
microscopic dynamical counterpart of thermodynamics also pointing to the
existence of some major change in the geometry of the mechanical manifolds at
the thermodynamical transition. Through the microcanonical definition of the
entropy, a relationship between thermodynamics and the extrinsic geometry of
the constant energy surfaces $\Sigma_E$ of phase space can be naturally
established. In this framework, an approximate formula is worked out,
determining a highly non-trivial relationship between temperature and topology
of the $\Sigma_E$. Whence it can be understood that the appearance of a phase
transition must be tightly related to a suitable major topology change of the
$\Sigma_E$. This contributes to the understanding of the origin of phase
transitions in the microcanonical ensemble.
