  In the framework of a recently developed theory for Hamiltonian chaos, which
makes use of the formulation of Newtonian dynamics in terms of Riemannian
differential geometry, we obtained analytic values of the largest Lyapunov
exponent for the Fermi-Pasta-Ulam-beta model (FPU-beta) by computing the time
averages of the metric tensor curvature and of its fluctuations along
analytically known unstable periodic orbits (UPOs). The agreement between our
results and the Lyapunov exponents obtained by means of standard numerical
simulations supports the fact that UPOs are reliable probes of a general
dynamical property as chaotic instability.
