  In this paper we consider the space of those probability distributions which
maximize the $q$-R\'enyi entropy. These distributions have the same parameter
space for every $q$, and in the $q=1$ case these are the normal distributions.
Some methods to endow this parameter space with Riemannian metric is presented:
the second derivative of the $q$-R\'enyi entropy, Tsallis-entropy and the
relative entropy give rise to a Riemannian metric, the Fisher-information
matrix is a natural Riemannian metric, and there are some geometrically
motivated metrics which were studied by Siegel, Calvo and Oller, Lovri\'c,
Min-Oo and Ruh. These metrics are different therefore our differential
geometrical calculations based on a unified metric, which covers all the above
mentioned metrics among others. We also compute the geometrical properties of
this metric, the equation of the geodesic line with some special solutions, the
Riemann and Ricci curvature tensors and scalar curvature. Using the
correspondence between the volume of the geodesic ball and the scalar curvature
we show how the parameter $q$ modulates the statistical distinguishability of
close points. We show that some frequently used metric in quantum information
geometry can be easily recovered from classical metrics.
