Geometry of Manifolds of Diffeomorphisms

Abstract

Our main goal in this chapter is to study the geometry of the infinite-dimensional manifold formed by the H^s-maps of a smooth compact manifold. Here s is a sufficiently large integer and H ^s denotes the Sobolev class. More specifically, our attention is primarily paid to the Hilbert manifold of H ^s - or C^k-diffeomorphisms. Then we use the notions of infinite-dimensional Riemannian geometry to describe the mathematical model of the flow of a barotropic fluid and diffuse matter as infinite-dimensional analogs of the mechanical systems of Chap. 2.