Jets of Sections
This subsection deals with the notion of jets of local sections a vector bundle. Almost everything in this section generalizes mutatis mutandis to the case of local sections of a fibre bundle, a Banach bundle over a Banach manifold, a fibered manifold, or quasi-coherent sheaves over schemes. Furthermore, these examples of possible generalizations are certainly not exhaustive.
Suppose that E is a finite-dimensional smooth vector bundle over a manifold M, with projection . Then sections of E are smooth functions such that is the identity automorphism of M. The jet of a section s over a neighborhood of a point p is just the jet of this smooth function from M to E at p.
The space of jets of sections at p is denoted by . Although this notation can lead to confusion with the more general jet spaces of functions between two manifolds, the context typically eliminates any such ambiguity.
Unlike jets of functions from a manifold to another manifold, the space of jets of sections at p carries the structure of a vector space inherited from the vector space structure on the sections themselves. As p varies over M, the jet spaces form a vector bundle over M, the k-th order jet bundle of E, denoted by Jk(E).
- Example: The first-order jet bundle of the tangent bundle.
- We work in local coordinates at a point. Consider a vector field
- in a neighborhood of p in M. The 1-jet of v is obtained by taking the first-order Taylor polynomial of the coefficients of the vector field:
- In the x coordinates, the 1-jet at a point can be identified with a list of real numbers . In the same way that a tangent vector at a point can be identified with the list (vi), subject to a certain transformation law under coordinate transitions, we have to know how the list is affected by a transition.
- So let us consider the transformation law in passing to another coordinate system yi. Let wk be the coefficients of the vector field v in the y coordinates. Then in the y coordinates, the 1-jet of v is a new list of real numbers . Since
- it follows that
- So
- Expanding by a Taylor series, we have
- Note that the transformation law is second order in the coordinate transition functions.
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