  We show how Fisher's information already known particular character as the
fundamental information geometric object which plays the role of a metric
tensor for a statistical differential manifold, can be derived in a relatively
easy manner through the direct application of a generalized logarithm and
exponential formalism to generalized information-entropy measures. We shall
first shortly describe how the generalization of information-entropy measures
naturally comes into being if this formalism is employed and recall how the
relation between all the information measures is best understood when described
in terms of a particular logarithmic Kolmogorov-Nagumo average. Subsequently,
extending Kullback-Leibler's relative entropy to all these measures defined on
a manifold of parametrized probability density functions, we obtain the metric
which turns out to be the Fisher information matrix elements times a real
multiplicative deformation parameter. The metrics independence from the
non-extensive character of the system, and its proportionality to the rate of
change of the multiplicity under a variation of the statistical probability
parameter space, emerges naturally in the frame of this representation.
