  We develop a geometric approach to operator growth and Krylov complexity in
many-body quantum systems governed by symmetries. We start by showing a direct
link between a unitary evolution with the Liouvillian and the displacement
operator of appropriate generalized coherent states. This connection maps
operator growth to a purely classical motion in phase space. The phase spaces
are endowed with a natural information metric. We show that, in this geometry,
operator growth is represented by geodesics and Krylov complexity is
proportional to a volume. This geometric perspective also provides two novel
avenues towards computation of Lanczos coefficients and sheds new light on the
origin of their maximal growth. We describe the general idea and analyze it in
explicit examples among which we reproduce known results from the
Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and
Heisenberg-Weyl symmetries, and generalize the discussion to conformal field
theories. Finally, we use techniques from quantum optics to study operator
evolution with quantum information tools such as entanglement and Renyi
entropies, negativity, fidelity, relative entropy and capacity of entanglement.
