Jet bundle

In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form.

Historically, jet bundles are attributed to Ehresmann, and were an advance on the method (prolongation ) of Elie Cartan, of dealing geometrically with higher derivatives , by imposing differential form conditions on newly-introduced formal variables. Jet bundles are sometimes called sprays.

Motivating example

Let be the trivial bundle over B. Then sections of this bundle are smooth maps . Two such maps f and g are said to be equivalent at y in B if

(here d(x,y) denotes distance in any fixed Riemannian metric on B). The classes of equivalence of such maps at y form the fiber of the first jet bundle at y.

The n-th jet bundle is constructed by repeating this operation n times.

What follows is a generalization of this construction to an arbitrary fiber bundle E.

Another motivation for the study of jet bundles is the need to explain the transformation properties of the Christoffel symbols under a change of coordinates. The Christoffel symbols do not transform as tensors on the tangent bundle; rather, they transform as tensors on the jet bundle.

Definition

Given a differential manifold B and a fiber bundle E over B which is also a differential manifold, the fiber F_x at a point x in B will also be a differential manifold. Hence for any point y in F_x the tangent space T_yF_x of F_x at y is a linear subspace of the full tangent space of E at y. T_yF_x is called the vertical subspace. The full tangent space can be decomposed into a direct sum of the vertical subspace and a complementary horizontal subspace. Now we can define a fiber bundle J over E whose fiber at a y is the set of all possible horizontal subspaces. Viewed as a fiber bundle over B, J is called the first order jet bundle over B.

The jet bundle of order n over B is now defined recursively as the first order jet bundle over the jet bundle of order n-1 over B.

Holonomic sections

Given a smooth section of the n-1st jet bundle, it induces a unique section of the n^th jet bundle by taking the horizontal subspace to be the tangent space to the section. Repeating this operation defines a unique section of the n^th jet bundle out of section of original bundle, called the n^th prolongation.

All sections which can be obtained this way are called holonomic.