The aim of computational anatomy is to develop models for understanding the physiology of organs and tissues. The diffeomorphic non-rigid registration is a validated instrument for the detection of anatomical changes on medical images and is based on a rich mathematical background. For instance, the ''large deformation diffeomoprhic metric mapping'' framework defines a Riemannian setting by providing an opportune right invariant metric on the tangent space, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. In alternative, stationary velocity fields have been proposed for the diffeomorphic registration based on the one-parameter subgroups from Lie groups theory. In spite of the higher computational efficiency, the geometrical setting of the latter method is more vague, especially regarding the relationship between one-parameter subgroups and geodesics. In this study, we present the relevant properties of the Lie groups for the definition of geometrical properties within the one-parameter subgroups parametrization, and we define the geometrical structure for computing geodesics and for parallel transporting. The theorethical results are applied to the image registration context, and discussed in light of the practical computational problems.