  We compute both analytically and numerically the geometry of the parameter
space of the anharmonic oscillator employing the quantum metric tensor and its
scalar curvature. A novel semiclassical treatment based on a Fourier
decomposition allows to construct classical analogues of the quantum metric
tensor and of the expectation values of the transition matrix elements. A
detailed comparison is presented between exact quantum numerical results, a
perturbative quantum description and the semiclassical analysis. They are shown
to coincide for both positive and negative quadratic potentials, where the
potential displays a double well. Although the perturbative method is unable to
describe the region where the quartic potential vanishes, it is remarkable that
both the perturbative and semiclassical formalisms recognize the negative
oscillator parameter at which the ground state starts to be delocalized in two
wells.
