Jon Herman, Pure Mathematics Department, University of Waterloo
“Symplectomorphisms and Geodesic Flow”
I will first define symplectic manifolds and the isomorphisms between them, called symplectomorphisms. Given any manifold, the cotangent bundle can always be given a canonical symplectic structure. Using this structure, one can provide an answer to the question of when a diffeomorphism between symplectic manifolds is a symplectomorphism. This will follow by considering a specific type of ”Lagrangian submanifold” and solving the Hamilton equations.
Lastly, I will discuss how symplectomorphisms can be used to interpret geodesic flow on a Riemannian manifold. In particular, after identifying the cotangent bundle with the tangent bundle, I will show how the Riemannian distance function generates a symplectomorphism that coincides with the exponential map.