Authors examine groups of diffeomorphisms on a smooth manifold. Their interest stems from V. Arnold's seminal 1956 paper on hydrodynamics in which he suggested that the Euler equation modeling a perfect fluid on an oriented Riemannian manifold can be reformulated as the equation for geodesics on the group of volume and orientation preserving the manifold's diffeomorphisms. There is no index. Annotation ©2013 Book News, Inc., Portland, OR (booknews.com)