  We present some perspectives in the construction of explicit cell structures
on real flag manifolds, equivariant with respect to the (free) action of the
Weyl group. Such structures could be obtained from Dirichlet-Voronoi
fundamental domains associated to these manifolds, defined using normal
homogeneous metrics.
  First, we motivate the study by reviewing the Riemannian geometric properties
of the flag manifold $\mathcal{F}_3(\mathbb{R})=O(3)/O(1)^3$ of
$SL_3(\mathbb{R})$ and exhibit some geodesic properties of an
$\mathfrak{S}_3$-equivariant cell structure of $\mathcal{F}_3(\mathbb{R})$
previously constructed by R. Chiriv\`i, M. Spreafico and the author. In
particular, the 1-cells are seen to be open geodesic arcs.
  Then, we define Dirichlet-Voronoi domains for Riemannian manifolds, equipped
with a finite group of isometries and give their first properties. Under a
rather strong condition on the injectivity radius of the manifold, such domains
are a reasonable starting point towards the construction of cell structures. We
prove moreover that a nice enough cell structure on such a domain induces an
equivariant cell structure on the whole manifold. We apply these considerations
to produce a new $\mathfrak{S}_3$-equivariant cell structure on
$\mathcal{F}_3(\mathbb{R})$.
