  The paper considers the Euler system of PDE on a smooth compact Riemannian
manifold of positive curvature without boundary, and the sphere
${\mathbb{S}}^2$ in particular. The paper interprets the Euler equations as a
transport problem for the fluid density under dynamics governed by the gradient
of the internal energy of the fluid. The paper develops the notion of transport
cost in the tangent bundle, and compares its properties with the Wasserstein
transportation cost on the manifold. There are applications to the discrete
approximation to the Euler equations in the style of Gangbo and Wesdickenberg
({\sl Comm. Partial Diff. Equations} {\bf 34} (2009), 1041-1073), except that
the analysis is heavily dependent upon the curvature of the underlying
manifold. The internal energy is assumed to satisfy convexity conditions that
allow analysis via $\Phi$-entropy entropy-production inequalities, and the
results apply to the power law $\rho^\gamma$ where $1<\gamma<3/2$, which
includes the case of a diatomic gas. The paper proves existence of weak
solutions of the continuity equation, and gives a sufficient condition for
existence of weak solutions to the acceleration equation.
