Let [math] be a measure space, and let [math] denote the set of the [math] -almost surely strictly positive probability densities. It was shown by Pistone and Sempi in 1995 that the global geometry on [math] can be realized by an affine atlas whose charts are defined locally by the mappings [math] , where [math] is a suitable open set containing [math] , [math] is the Kullback--Leibler relative information and [math] is the vector space of centred and exponentially [math] -integrable random variables. In the present paper we study the transformation of such an atlas and the related manifold structure under basic transformations, i.e. measurable transformation of the sample space. A generalization of the mixed parametrization method for exponential models is also presented.