  We consider the problem of existence of certain symmetrical solutions of
Stokes equation on a three-dimensional manifold $M$ with a general metric
possessing symmetry. These solutions correspond to unidirectional flows. We
have been able to determine necessary and sufficient conditions for their
existence. Symmetric unidirectional flows are fundamental for deducing the
so-called Darcy's law, which is the law governing fluid flow in a Hele-Shaw
cell embedded in the environment $M$. Our main interest is to depart from the
usual, flat background environment, and consider the possibility of an
environment of arbitrary constant curvature $K$ in which a cell is embedded. We
generalize Darcy's law for particular models of such spaces obtained from
$\real^3$ with a conformal metric. We employ the calculus of differential forms
for a simpler and more elegant approach to the problems herein discussed.
