  This paper is a strongly geometrical approach to the Fisher distance, which
is a measure of dissimilarity between two probability distribution functions.
The Fisher distance, as well as other divergence measures, are also used in
many applications to establish a proper data average. The main purpose is to
widen the range of possible interpretations and relations of the Fisher
distance and its associated geometry for the prospective applications. It
focuses on statistical models of the normal probability distribution functions
and takes advantage of the connection with the classical hyperbolic geometry to
derive closed forms for the Fisher distance in several cases. Connections with
the well-known Kullback-Leibler divergence measure are also devised.
