Marsden, J. E., D. G. Ebin, and A. E. Fischer
Parts I and II form a unit. Only a small amount of material form Part I is needed in Part III--an acquaintance with the rudiments of the diffeomorphism groups. The sort of global analysis used in hydrodynamics is developed in Part I. The machinery needed in relativity--infinite dimensional Hamiltonian systems--is developed as it is used. This organization should make it possible for one to read Part III separately if desired.
Because of inevitable time restrictions, it was necessary to make some selection with regard to the topics and their depth of discussion. There was an effort made to include topics and points of view that would be of interest to geometers and global analysts. Nevertheless, we hope the overall perspective presented gives a reasonably correct picture of some questions of interest to specialists in theoretical hydrodynamics and general relativity.
As far as prerequisites go, this article assumes the following: A knowledge of the general facts about manifolds, differenial forms and Riemannian geometry. For example, Abraham, Marsden, and Ratiu [1988] and Bishop and Crittenden [1964] contain more than enough background. We have included a certain amount of introductory material in fluid mechanics in §2. One can supplement this by consulting standard texts. For general realtivity we assume, in addition to the geometry above, some familiarity with the basic ideas of special and general relativity, as may be found in the book of Misner, Thorne and Wheeler.
Much of the material on hydrodynamics is taken from Ebin and Marsden [1970]. However, our exposition here is more informal and gets at several points from a different direction. The exposition regarding turbulence is largely influenced by the work of Ruelle and Takens, while the material on relativity is mostly taken from Fischer and Marsden [1972].