A distance between multivariate normal distributions based in an embedding into the siegel group

This paper shows an embedding of the manifold of multivariate normal densities with informative geometry into the manifold of definite positive matrices with the Siegel metric. This embedding allows us to obtain a general lower bound for the Rao distance, which is itself a distance, and we suggest employing it for statistical purposes, taking into account the similitude of the above related metrics. Further-more, through this embedding, general statistical tests of hypothesis are derived, and some geometrical properties are studied too.