Let be a surjective submersion of differential manifolds, where ( means real analytic). For all integers with , we will define a fibre bundle over , called the -th jet bundle of over . The fibre of this bundle above a point can be interpreted as the set of equivalence classes of local sections of , where two sections are considered equivalent if their first derivatives at are equal. The equivalence class of a section is then the jet of that section; it indicates the direction of the section locally at . This concept has much in common with that of the germ of a smooth function on a manifold: it not only the value of a function at a point, but also some about the behaviour of the function near that point.
We will now define each jet bundle of over as a set with a projection map to , and we describe the concept of prolongation of sections. After that, we give a slightly different construction allowing us to put a manifold structure on each of the jet bundles.
For every open subset of , we denote by the set of sections of over , i.e. the set of functions such that . Every point of has an open subset such that there exists at least one section of over , due to the assumption that is a surjective submersion. For all , we define the fibre of above by
where the equivalence relation is defined by and induce the same map between the fibres at and of the -th iterated tangent bundles of and , respectively. (Note that the fibres in the -th iteration are the same if and only if the induced maps are already the same in the -st iteration). We will denote the equivalence class of a pair by . As a set, is defined as the disjoint union of the sets with . Write for the ‘obvious’ projection map, defined by
Notice that is just itself.
Suppose we have some section of over an open subset of . By sending every point to the equivalence class we obtain a section of over for each , called the -th prolongation of . Composing this section on the left with gives back the original section of .
0.3 The bundle structure
Instead of defining all the jet bundles at once, we may choose to define only the first jet bundle in the way described above. After equipping the first jet bundle with the structure of a differential manifold, which we will do below, we can then inductively define as the first jet bundle of over for . This is useful because the manifold structure only needs to be defined for .
We make into an affine bundle over , locally trivial of rank , in the following way. We cover with charts , where is a diffeomorphism between open subsets and . Without loss of generality, we assume that is contained in the domain of a chart on , with . Here and are the local dimensions of and , respectively.
The definition of the equivalence relation on means that the association is well-defined and injective for each . The image of consists of the matrices with the property that multiplying them on the left with the matrix corresponding to the tangent map gives the identity matrix. These matrices form a -dimensional linear subspace , and is a submanifold of .
We both the differentiable structure of and a local trivialisation of as a vector bundle by requiring that
be a diffeomorphism and an -linear map. Since the effect of a change of charts on or is multiplying each by matrices depending differentiably on (namely, the derivatives of the glueing maps), this gives a well-defined vector bundle structure on all of .
Iterating the above construction by defining as the first jet bundle of over , each jet bundle becomes a vector bundle over and a fibre bundle over . Normally, only is a vector bundle over .
|Date of creation||2013-03-22 15:28:42|
|Last modified on||2013-03-22 15:28:42|
|Last modified by||rspuzio (6075)|