Pseudo-Manifold Geometries with Applications

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969), i.e., an axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only
invalided but in multiple distinct ways and a Smarandache n-manifold is a n-manifold that support a Smarandache geometry.