Jet bundle Guide, Meaning , Facts, Information and Description

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 possible to write differential equations on sectionss 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.

Table of contents

1 Motivating example 2 Definition 3 Holonomic sections

Motivating example

Let be a trivial bundle over , then sections of this bundle can be described by smooth maps . Two such maps are said to be equivalent at p if

(here denotes distance in any fixed Riemannian metric on B). The classes of equivalences of such maps at p form the fiber of first jet bundle at p.

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.

Definition

Given a differential manifold B and a fiber bundle E over it which is also a differential manifold, which means the fiber F_x at a point x on B is also a differential manifold. So, for any point y on E, it belongs to a fiber which also happens to be a submanifold and so, the tangent space at y of the fiber it belongs to is a subspace of the tangent space at y of E. The former space is called the vertical subspace. The full tangent space decomposes into a direct sum of a vertical subspace and another subspace called the horizontal subspace. However, the horizontal subspace is most definitely nonunique. We can now define a fiber bundle over E whose fiber at a point y on E is the set of all possible horizontal subspaces. Now, any fiber bundle over E is also a fiber bundle over B under the composition of projections. This space, when viewed as fiber bundle over B is called a first order jet bundle over B.

The n^th order jet bundle over B is defined recursively as follows. Suppose we have already defined the n-1^th order jet bundle. Then, the first order jet bundle associated with the n-1^th order jet bundle is the n^th order jet bundle.

Holonomic sections

Given a smooth section of n-1^th jet bundle it induce a unique section of n^th jet bundlejet bundle by taking horizontal subspace to be the tangent space to the section. Repeating this operation defines unique section of n^th jet bundle out of section of original bundle. All sections which can be tained this way are called holonomic.

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