  The study of minimal surfaces has a long history, due to the important
applications. Given a fixed boundary, one wants to minimise the surface area:
this can be used, for example, to minimise the area of the roof of a building.
Similarly, looking for constant mean curvature (CMC) provides us with many
interesting applications in physics: one of the easiest examples are soap
bubbles. In this work however we occupy ourselves with minimal and constant
mean curvature surfaces in the three-dimensional sphere $S^3$ and its dual
space $\Sigma^3$.
  In Chapter 1 we give a brief overview of the tools of Riemannian and
Lorentzian geometry that we will use. We then take a closer look at $S^3$,
computing its Levi-Civita connection and sectional curvatures: in Chapter 2
with respect to the Riemannian metric g and in Chapter 4 with respect to the
Lorentzian metric h. Further, we determine some minimal and CMC tori inside
($S^3$, g) in Chapter 3 and in ($S^3$, h) in Chapter 5.
  We then proceed with the dual space $\Sigma^3$ of $S^3$. In Chapter 6, we
calculate the Levi-Civita connection and sectional curvatures with respect to
g, and with respect to h in Chapter 8. Again we look for minimal and CMC tori
of a certain family in ($\Sigma^3$, g) in Chapter 7 and in ($\Sigma^3$, h) in
Chapter 9.
  In the appendix, the reader will find a Maple program. It was written to
check the computations of the $S^3$ cases, but it can easily be adapted
to$\Sigma^3$.

Differential Geometry