Generalized Swan’s theorem and its application

Abstract

Swan’s theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define A ( r ) {A^{(r)}} -maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan’s theorem and thus generalize Swan’s theorem to include non-linear differential operators as morphisms. An A ( r ) {A^{(r)}} -manifold structure is introduced on the space of sections of a fiber bundle through charts with A ( r ) {A^{(r)}} -maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of A ( r ) {A^{(r)}} -maps.