jet bundle


0.1 Introduction

Let p:Xβ†’B be a surjectivePlanetmathPlanetmath submersionMathworldPlanetmath of π’žr differential manifolds, where r∈{0,1,…}βˆͺ{∞,Ο‰} (π’žΟ‰ means real analytic). For all integers kβ‰₯0 with k≀r, we will define a fibre bundle JBk⁒X over X, called the k-th jet bundleMathworldPlanetmath of X over B. The fibre of this bundle above a point x∈X can be interpreted as the set of equivalence classesMathworldPlanetmathPlanetmath of local sections of p x, where two sectionsPlanetmathPlanetmath are considered equivalentMathworldPlanetmathPlanetmathPlanetmath if their first k derivativesPlanetmathPlanetmath at x are equal. The equivalence class of a section is then the jet of that section; it indicates the direction of the section locally at x. This concept has much in common with that of the germ of a smooth functionMathworldPlanetmath 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.

0.2 Construction

We will now define each jet bundle of X over B as a set with a projection map to X, and we describe the concept of prolongation of sections. After that, we give a slightly different construction allowing us to put a manifold structureMathworldPlanetmath on each of the jet bundles.

For every open subset of B, we denote by Γ⁒(U,X) the set of sections of p over U, i.e.Β the set of π’žr functions s:Uβ†’p-1⁒U such that p|p-1⁒U∘s=idU. Every point of B has an open subset U such that there exists at least one section of p over U, due to the assumptionPlanetmathPlanetmath that p is a surjective submersion. For all x∈X, we define the fibre of JBk⁒X above x by

JBk⁒X⁒(x)={(U,s):UβŠ‚B⁒ open,x∈U,sβˆˆΞ“β’(U,X),s⁒(p⁒(x))=x}/∼,

where the equivalence relation ∼ is defined by s∼s′⇔s and sβ€² induce the same map between the fibres at p⁒(x) and x of the k-th iterated tangent bundlesMathworldPlanetmath of B and X, respectively. (Note that the fibres in the k-th iteration are the same if and only if the induced maps are already the same in the (k-1)-st iteration). We will denote the equivalence class of a pair (U,s) by [U,s]. As a set, JBk⁒X is defined as the disjoint unionMathworldPlanetmath of the sets JBk⁒X⁒(x) with x∈X. Write Ο€:JBk⁒Xβ†’X for the β€˜obvious’ projection map, defined by

π⁒([U,s])=x⁒ for ⁒[U,s]∈JBk⁒X⁒(x).

Notice that JB0⁒X is just X itself.

Suppose we have some section s of X over an open subset U of B. By sending every point y∈U to the equivalence class [U,s]∈JBk⁒X⁒(s⁒(y)) we obtain a section of Ο€βˆ˜p:JBk⁒Xβ†’B over U for each kβ‰₯0, called the k-th prolongation of s. Composing this section on the left with Ο€ gives back the original section s of X.

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 JBk+1⁒X as the first jet bundle of JBk⁒X over B for kβ‰₯1. This is useful because the manifold structure only needs to be defined for JB1⁒X.

We make JB1⁒X into an affine bundle over X, locally trivial of rank dim⁑B⁒(dim⁑X-dim⁑B), in the following way. We cover X with charts (W,Ο•,Wβ€²), where Ο• is a diffeomorphism between open subsets WβŠ‚X and Wβ€²βŠ‚β„n. Without loss of generality, we assume that p⁒(W) is contained in the domain V of a chart (V,ψ,Vβ€²) on B, with Vβ€²βŠ‚β„m. Here n and m are the local dimensionsPlanetmathPlanetmath of X and B, respectively.

For all x∈W and all [U,s]∈JB1⁒X⁒(x), we have the tangent mapMathworldPlanetmath T⁒s⁒(p⁒(x)), which is a linear map from T⁒V⁒(p⁒(x)) to T⁒W⁒(x). These tangent spacesPlanetmathPlanetmath are isomorphic to ℝm and ℝn via the chosen charts, so that T⁒s⁒(p⁒(x)) acts as a matrix Mx⁒([U,s])βˆˆβ„nΓ—m:

\xymatrix⁒T⁒W⁒(x)⁒\ar⁒[r]≃⁒&⁒ℝn⁒T⁒V⁒(p⁒(x))⁒\ar⁒[u]T⁒s⁒(p⁒(x))⁒\ar⁒[r]≃⁒&⁒ℝm⁒\ar⁒[u]Mx⁒([U,s])

The definition of the equivalence relation ∼ on JB1⁒X⁒(x) means that the association [U,s]↦Mx⁒([U,s]) is well-defined and injectivePlanetmathPlanetmath for each x∈W. The image of Mx consists of the matrices with the property that multiplying them on the left with the mΓ—n matrix corresponding to the tangent map T⁒p⁒(x) gives the mΓ—m identity matrixMathworldPlanetmath. These matrices form a m⁒(n-m)-dimensional linear subspace LxβŠ‚β„nΓ—m, and L=⋃x∈W{x}Γ—Lx is a submanifoldMathworldPlanetmath of W×ℝnΓ—m.

We both the differentiable structure of Ο€-1⁒U and a local trivialisation of JB1⁒X as a vector bundle by requiring that

Ο€-1⁒W β†’ L
[U,s] ↦ (π⁒([U,s]),Mπ⁒([U,s])⁒([U,s]))

be a diffeomorphism and an ℝ-linear map. Since the effect of a change of charts on W or V is multiplying each Mx⁒([U,s]) by matrices depending differentiably on x∈W (namely, the derivatives of the glueing maps), this gives a well-defined vector bundle structure on all of JB1⁒X.

Iterating the above construction by defining JBk+1⁒X as the first jet bundle of JBk⁒X over B, each jet bundle JBk+1⁒X becomes a vector bundle over JBk⁒X and a fibre bundle over X. Normally, only JB1⁒X is a vector bundle over X.

Title jet bundle
Canonical name JetBundle
Date of creation 2013-03-22 15:28:42
Last modified on 2013-03-22 15:28:42
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 58A20
Defines jet