  In this thesis we prove that the homogeneous incompressible Euler equation of
hydrodynamics on the Sobolev spaces $H^s(\R^n)$, $n \geq 2$ and $s > n/2+1$,
can be expressed as a geodesic equation on an infinite dimensional manifold. As
an application of this geometric formulation we prove that the solution map of
the incompressible Euler equation, associating intial data in $H^s(\R^n)$ to
the corresponding solution at time $t > 0$, is nowhere locally uniformly
continuous and nowhere differentiable.
