Jets and Natural Bundles

In this chapter we start our systematic treatment of geometric objects and operators. It has become commonplace to think of geometric objects on a manifold M as forming fiber bundles over the base M and to work with sections of these bundles. The concrete objects were frequently described in coordinates by their behavior under the coordinate changes. Stressing the change of coordinates, we can say that local diffeomorphisms on the base manifold operate on the bundles of geometric objects. Since a further usual assumption is that the resulting transformations depend only on germs of the underlying morphisms, we actually deal with functors defined on all open submanifolds of M and local diffeomorphisms between them (let us recall that local diffeomorphisms are globally defined locally invertible maps), see the preface. This is the point of view introduced by [Nijenhuis, 72] and worked out later by [Terng, 78], [Palais, Terng, 77], [Epstein, Thurston, 79] and others. These functors are fully determined by their restriction to any open submanifold and therefore they extend to the whole category .M f m of m-dimensional manifolds and local diffeomorphisms. An important advantage of such a definition of bundles of geometric objects is that we get a precise definition of geometric operators in the concept of natural operators. These are rules transforming sections of one natural bundle into sections of another one and commuting with the induced actions of local diffeomorphisms between the base manifolds.