  We study the Killing vectors of the quantum ground-state manifold of a
parameter-dependent Hamiltonian. We find that the manifold may have symmetries
that are not visible at the level of the Hamiltonian and that different quantum
phases of matter exhibit different symmetries. We propose a Bianchi-based
classification of the various ground-state manifolds using the Lie algebra of
the Killing vector fields. Moreover, we explain how to exploit these symmetries
to find geodesics and explore their behaviour when crossing critical lines. We
briefly discuss the relation between geodesics, energy fluctuations and
adiabatic preparation protocols. Our primary example is the anisotropic
transverse-field Ising model. We also analyze the Ising limit and find analytic
solutions to the geodesic equations for both cases.
