Since the seminal work of Arnold on the Euler equations (1966), many equations in hydrodynamics were shown to be geodesic equations of diffeomorphism groups of manifolds, with respect to various Sobolev norms. This led to new ways to study these PDEs, and also initiated the study of of the geometry of those groups as (infinite dimensional) Riemannian manifolds.
In this talk I will introduce the Riemannian structure induced on the diffeomorphism group by different Sobolev norms, and will focus on one question -- is the geodesic distance between different diffeomorphisms positive or not? While the answer is always positive for finite dimensional Riemannian manifolds, we will see that for the diffeomorphism group the answer strongly depends on the norm.
Based on a joint work with Robert Jerrard.