Classical differential equations on manifolds

Abstract

1. Types of differential objects on manifolds. Let D be a differential object, say, the Laplace potential partial differential equation, the wave equation, the diffusion equation, or one of the corresponding differential operators defined in the Cartesian coordinates of Euclidean n-space Rn. We shall say that D is well-defined on a differentiable manifold Mn (connected, separable metric space with C* local coordinates) in case there is a differential equation or a differential operator defined on Mn, which in a certain atlas (covering of Mn by a subcollection of the differentiable coordinate systems) is expressed by D. The collection of all differentiable homeomorphisms of open sets of Rn into R , which preserve D, is the pseudogroup of D, cf. [II]. Then we can state that Mn admits the type of differential object D if and only if Mn has an atlas whose coordinate transition maps belong to the pseudogroup of D. In this paper we shall find topological and geometrical properties of manifolds which admit the classical differential objects listed above. In the next section we relate this problem to the more standard one of reducing the structure group of the principal bundle of bases B(Mn). In section three we study applications of the general theory to the Laplace and wave partial differential equations and also to linear ordinary differential equations with constant coefficients. Finally, in the appendix we state properties of bundle reductions to totally disconnected subgroups; these results being immediate generalizations of an earlier work of the authors [5] and of relevance for the present study. 2. Bundle reduction for the potential, wave, diffusion equations and operators. We note that the pseudogroup for the potential operator