A paradoxist geometry focuses attention on the parallel postulate, the same postulate of Euclid that Gauss, Bolyai, Lobachevski, and Riemann sought to contradict. In fact, Riemann began the study of geometric spaces that are non-uniform with respect to the parallel postulate, since in a Riemannian manifold, the curvature may change from point to point. This corresponds roughly with what we will call semi-paradoxist. It would seem, therefore, that a study of Smarandache geometry should start with Riemannian manifolds, and inadvertently, it has. Unfortunately, describing and manipulating Riemannian manifolds is far from trivial, and many Smarandache-type structures probably cannot exist in a Riemannian manifold.