  This paper tackles Hamiltonian chaos by means of elementary tools of
Riemannian geometry. More precisely, a Hamiltonian flow is identified with a
geodesic flow on configuration space-time endowed with a suitable metric due to
Eisenhart. Until now, this framework has never been given attention to describe
chaotic dynamics. A gap that is filled in the present work. In a
Riemannian-geometric context, the stability/instability of the dynamics depends
on the curvature properties of the ambient manifold and is investigated by
means of the Jacobi--Levi-Civita (JLC) equation for geodesic spread. It is
confirmed that the dominant mechanism at the ground of chaotic dynamics is
parametric instability due to curvature variations along the geodesics. A
comparison is reported of the outcomes of the JLC equation written also for the
Jacobi metric on configuration space and for another metric due to Eisenhart on
an extended configuration space-time. This has been applied to the
H\'enon-Heiles model, a two-degrees of freedom system. Then the study has been
extended to the 1D classical Heisenberg XY model at a large number of degrees
of freedom. Both the advantages and drawbacks of this geometrization of
Hamiltonian dynamics are discussed. Finally, a quick hint is put forward
concerning the possible extension of the differential-geometric investigation
of chaos in generic dynamical systems, including dissipative ones, by resorting
to Finsler manifolds.
