  We study the fuzzy spaces (as special examples of noncommutative manifolds)
with their quasicoherent states in order to find their pertinent metrics. We
show that they are naturally endowed with two natural "quantum metrics" which
are associated with quantum fluctuations of "paths". The first one provides the
length the mean path whereas the second one provides the average length of the
fluctuated paths. Onto the classical manifold associated with the quasicoherent
state (manifold of the mean values of the coordinate observables in the state
minimising their quantum uncertainties) these two metrics provides two
minimising geodesic equations. Moreover, fuzzy spaces being not torsion free,
we have also two different autoparallel geodesic equations associated with two
different adiabatic regimes in the move of a probe onto the fuzzy space. We
apply these mathematical results to quantum gravity in BFSS matrix models, and
to the quantum information theory of a controlled qubit submitted to noises of
a large quantum environment.
