Abstract In the past few years there has been a growing interest on geometric frameworks to learn supervised classififi- cation models on Riemannian manifolds [31, 27]. A popular framework, valid over any Riemannian manifold, was proposed in [31] for binary classifification. Once moving from binary to multi-class classifification this paradigm is not valid anymore, due to the spread of multiple positive classes on the manifold [27]. It is then natural to ask whether the multi-class paradigm could be extended to operate on a large class of Riemannian manifolds. We propose a mathematically well-founded classifification paradigm that allows to extend the work in [31] to multi-class models, taking into account the structure of the space. The idea is to project all the data from the manifold onto an affifine tangent space at a particular point. To mitigate the distortion induced by local diffeomorphisms, we introduce for the fifirst time in the computer vision community a well-founded mathematical concept, so-called Rolling map [21, 16]. The novelty in this alternate school of thought is that the manifold will be fifirstly rolled (without slipping or twisting) as a rigid body, then the given data is unwrapped onto the affifine tangent space, where the classifification is performed.