  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 $\lambda$ 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 $\lambda \to 0$. 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 $\lambda$ 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 $\lambda$, but for the ground state the total entropy is
minimised when $\lambda$ vanishes, and the corresponding uncertainty relation
is always fulfilled.
