The so-called Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined
on a radially symmetric space with non-constant negative curvature. This oscillator can be interpreted
as a smooth (super)integrable deformation of the usual N-dimensional harmonic oscillator in terms of
a non-negative parameter λ which is directly related to the curvature of the underlying space. In this
paper, a detailed study of the Shannon information entropy for the quantum version of the Darboux
III oscillator is presented, and the interplay between entropy and curvature is analysed. In particular,
analytical results for the Shannon entropy in the position space can be found in the N-dimensional case,
and the known results for the quantum states of the N-dimensional harmonic oscillator are recovered
in the limit of vanishing curvature λ → 0. However, the Fourier transform of the Darboux III wave
functions cannot be computed in exact form, thus preventing the analytical study of the information
entropy in momentum space. Nevertheless, we have computed the latter numerically both in the one
and three-dimensional cases and we have found that by increasing the absolute value of the negative
curvature (through a larger λ parameter) the information entropy in position space increases, while in
momentum space it becomes smaller. This result is indeed consistent with the spreading properties
of the wave functions of this quantum nonlinear oscillator, which are explicitly shown. The sum of
the entropies in position and momentum spaces has been also analysed in terms of the curvature: for
all excited states such total entropy decreases with λ, but for the ground state the total entropy is
minimized when λ vanishes, and the corresponding uncertainty relation is always fulfilled.
