  In this paper we give a Clifford bundle motivated approach to the wave
equation of a free spin $1/2$ fermion in the de Sitter manifold, a brane with
topology $M=\mathrm{S0}(4,1)/\mathrm{S0}(3,1)$ living in the bulk spacetime
$\mathbb{R}^{4,1}=(\mathring{M}=\mathbb{R}^{5},\boldsymbol{\mathring{g}})$ and
equipped with a metric field
$\boldsymbol{g:=-i}^{\ast}\boldsymbol{\mathring{g}%}$ with
$\boldsymbol{i}:M\rightarrow\mathring{M}$ being the inclusion map. To obtain
the analog of Dirac equation in Minkowski spacetime in the structure
$\mathring{M}$ we appropriately factorize the two Casimir invariants $C_{1}$
and $C_{2}$ of the Lie algebra of the de Sitter group using the constraint
given in the linearization of $C_{2}$ as input to linearize $C_{1}$. In this
way we obtain an equation that we called \textbf{DHESS1,}which in previous
studies by other authors was simply postulated.$.$Next we derive a wave
equation (called \textbf{DHESS2}) for a free spin $1/2$ fermion in the de
Sitter manifold using a heuristic argument which is an obvious generalization
of a heuristic argument (described in detail in Appendix D) permitting a
derivation of the Dirac equation in Minkowski spacetime and which shows that
such famous equation express nothing more than the fact that the momentum of a
free particle is a constant vector field over timelike \ integral curves of a
given velocity field. It is a remarkable fact that \textbf{DHESS1}and
\textbf{DHESS2}\ coincide. One of the main ingredients in our paper is the use
of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this
concept and its relation with covariant Dirac spinor fields usualy used by
physicists.
