  We consider the covariant Lyapunov vectors (CLV) of a high-dimensional
Hamiltonian flow in the case of long range potential, namely the Hamiltonian
Mean Field (HMF) problem, by studying the behavior of the Lyapunov spectra and
the Oseledets' splitting principal angles (the mutual orientation between
stable and unstable subspaces) when a phase transition takes place. Motivated
by several results connecting the dynamical properties of systems to their
Lyapunov exponents and vectors, we first find confirmation of an explicit
sensitivity of such quantities to the transition: our main finding is that the
mutual orientation between stable and unstable subspaces fluctuates in time
around specific constant values. Moreover, the fluctuations statistics are
different above and below the critical threshold, and thus intimately connected
to the transition. Pictorially, the evolution of the stability subspaces along
a reference orbits turns out to happen only through rigid global rotations plus
weak internal deformations between the two subspaces; we are not aware of any
theoretical explanation for such a non- trivial setting but, through analogies
with integrable systems, we provide some little insight on 'how much' the
system resembles an integrable one.
