They seem superficially similar in some ways
1) both have a given two tensor on the tangent space
2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical mapping from the tangent space to the cotangent space
3) they both give a canonical flow of the manifold and we can interpretate hamiltonian flow as geodesic flow
But the worlds are COMPLETELY different
1) riemannian manifolds have lots of local invariants, symplectic manifolds all look locally the same
2) it is always possible to put a riemannian tensor on a given manifold, symplectic manifolds are much rarer
Riemannian geometry provides a reasonable measure of length, symplectic geometry provides a reasonable definition of area. Why do these lead to such different theories?