  We show that the degenerate special Lagrangian equation, recently introduced
by Rubinstein-Solomon, induces a global equation on every Riemannian manifold,
and that for certain associated geometries this equation governs, as it does in
the Euclidean setting, geodesics in the space of positive Lagrangians. For
example, geodesics in the space of positive Lagrangian sections of a smooth
Calabi-Yau torus fibration are governed by the Riemannian DSL on the base
manifold. We then develop their analytic techniques, specifically modifications
of the Dirichlet duality theory of Harvey-Lawson, in the Riemannian setting to
obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and
hence continuous geodesics in the space of positive Lagrangians.
