The Algebra of linear differential operators on a vector bundle

Abstract

It is a well-known fact that, if the rings of continuous functions on some compact spaces X and Y are isomorphic, then the topological spaces X and Y are homeomorphic. Chapter III of the present work extends the theory of function rings to algebras of linear differential operators on a C∞ vector bundle; it will be proved, for instance, that if ξ and η are two real C∞ vector bundles over two paracompact manifolds M and N respectively such that their algebras of linear differential operators of order 0, 0po (ξ, ξ) and 0po (η, η) are isomorphic, then their base spaces M and N are C∞ diffeomorphic. The aim of chapters I and II is to introduce some basic concepts of differential geometry (manifolds, vector bundles,...) which will be used in the latter part; here the exposition strives towards some generality in that: 1) it avoids imposing finiteness conditions, where possible; 2) it attempts to pinpoint the exact nature of the mathematical objects involved, i.e. a vector field will not be defined as a mapping which satisfies certain properties but as a section of a tangent bundle.