Curves and surfaces in the three dimensional sphere placed in the space of quaternions

In this article we will show how to use Mathematica in dealing with curves and surfaces in the three dimensional unit sphere S-3 embedded in the four dimensional Euclidian space E-4. Since S-3 is the Lie group of unit quaternions and at the same time it is a space of constant curvature, the analogy of the theory of curves in E-3 holds. We calculate curvature and torsion of curves in S-3 by Mathematica. The Gauss map v of a surface in E-4 is decomposed into the two maps V+ and v(-). If the surface is contained in S-3, we can define another Gauss map v(s). We use Mathematica to visualize the shapes of the images of these Gauss maps. Finally, the meaning of these images becomes clear through the notion of the slant surface.