  We analyse the Kitaev honeycomb model, by means of the Berry curvature with
respect to Hamiltonian parameters. We concentrate on the ground-state
vortex-free sector, which allows us to exploit an appropriate Fermionisation
technique. The parameter space includes a time-reversal breaking term which
provides an analytical headway to study the curvature in phases in which it
would otherwise vanish. The curvature is then analysed in the limit in which
the time-reversal-symmetry-breaking perturbation vanishes. This provides
remarkable information about the topological phase transitions of the model. A
non-critical behaviour is found in the Berry curvature itself, which shows a
distinctive behaviour in the different phases. The analysis of the first
derivative shows a critical behaviour around the transition point.
