Jet bundle

In differential topology, the jet bundle is a certain construction that 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. Jets may also be seen as the coordinate free versions of Taylor expansions.
Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing "geometrically" with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.)
Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach.
Jets.
Suppose "M" is an "m"-dimensional manifold and that ("E", π, "M") is a fiber bundle. For "p" ∈ "M", let Γ(p) denote the set of all local sections whose domain contains "p". Let be a multi-index (an "m"-tuple of integers, not necessarily in ascending order), then define:
Define the local sections σ, η ∈ Γ(p) to have the same "r"-jet at "p" if
The relation that two maps have the same "r"-jet is an equivalence relation. An "r"-jet is an equivalence class under this relation, and the "r"-jet with representative σ is denoted formula_3. The integer "r" is also called the order of the jet, "p" is its source and σ("p") is its target.
Jet manifolds.
The "r"-th jet manifold of π is the set
We may define projections "πr" and "π""r",0 called the source and target projections respectively, by
If 1 ≤ "k" ≤ "r", then the "k"-jet projection is the function "πr,k" defined by
From this definition, it is clear that "πr" = "π" "π""r",0 and that if 0 ≤ "m" ≤ "k", then "πr,m" = "πk,m" "πr,k". It is conventional to regard "πr,r" as the identity map on "J r"("π") and to identify "J" 0("π") with "E".
The functions "πr,k", "π""r",0 and "πr" are smooth surjective submersions.
A coordinate system on "E" will generate a coordinate system on "J r"("π"). Let ("U", "u") be an adapted coordinate chart on "E", where "u" = ("xi", "uα"). The induced coordinate chart ("Ur", "ur") on "J r"("π") is defined by
where
and the formula_9 functions known as the derivative coordinates:
Given an atlas of adapted charts ("U", "u") on "E", the corresponding collection of charts ("U r", "u r") is a finite-dimensional "C"∞ atlas on "J r"("π").
Jet bundles.
Since the atlas on each "Jr(π)" defines a manifold, the triples "(Jr(π), πr,k, Jk(π))", "(Jr(π), πr,0, E)" and "(Jr(π), πr, M)" all define fibered manifolds. In particular, if "(E, π, M)" is a fiber bundle, the triple "(Jr(π), πr, M)" defines the "r"-th jet bundle of π.
If "W" ⊂ "M" is an open submanifold, then
If "p" ∈ "M", then the fiber formula_12 is denoted formula_13.
Let σ be a local section of π with domain "W" ⊂ "M". The "r"-th jet prolongation of σ is the map "jrσ": "W" → "Jr(π)" defined by
Note that π"r" "jrσ" = id"W", so "jrσ" really is a section. In local coordinates, "jrσ" is given by
We identify "j0σ" with σ.
Algebraic-geometric perspective.
An independently motivated construction of the sheaf of sections formula_16" is given"."
Consider a diagonal map formula_17, where the smooth manifold formula_18 is a locally ringed space by formula_19 for each open formula_20. Let formula_21 be the ideal sheaf of formula_22, equivalently let formula_21 be the sheaf of smooth germs which vanish on formula_22 for all formula_25. The pullback of the quotient sheaf formula_26 from formula_27 to formula_18 by formula_29 is the sheaf of k-jets.
The direct limit of the sequence of injections given by the canonical inclusions formula_30 of sheaves, gives rise to the infinite jet sheaf formula_31. Observe that by the direct limit construction it is a filtered ring.
Example.
If π is the trivial bundle ("M" × R, pr1, "M"), then there is a canonical diffeomorphism between the first jet bundle "J1(π)" and "T*M" × R. To construct this diffeomorphism, for each σ in Γ"M"(π) write formula_32.
Then, whenever "p" ∈ "M"
Consequently, the mapping
is well-defined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if "(xi, u)" are coordinates on "M" × R, where "u" = idR is the identity coordinate, then the derivative coordinates "ui" on "J1(π)" correspond to the coordinates ∂"i" on "T*M".
Likewise, if π is the trivial bundle (R × "M", pr1, R), then there exists a canonical diffeomorphism between "J1(π)" and R × "TM".
Contact structure.
The space "Jr"(π) carries a natural distribution, that is, a sub-bundle of the tangent bundle "TJr"(π)), called the "Cartan distribution". The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form "jrφ" for "φ" a section of π.
The annihilator of the Cartan distribution is a space of differential one-forms called contact forms, on "Jr"(π). The space of differential one-forms on "Jr"(π) is denoted by formula_35 and the space of contact forms is denoted by formula_36. A one form is a contact form provided its pullback along every prolongation is zero. In other words, formula_37 is a contact form if and only if
for all local sections σ of π over "M".
The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations. The Cartan distributions are completely non-integrable. In particular, they are not involutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jets "J∞" the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold "M".
Example.
Consider the case "(E, π, M)", where "E" ≃ R2 and "M" ≃ R. Then, "(J1(π), π, M)" defines the first jet bundle, and may be coordinated by "(x, u, u1)", where
for all "p" ∈ "M" and σ in Γ"p"(π). A general 1-form on "J1(π)" takes the form
A section σ in Γ"p"(π) has first prolongation
Hence, "(j1σ)*θ" can be calculated as
This will vanish for all sections σ if and only if "c" = 0 and "a" = −"bσ′(x)". Hence, θ = "b(x, u, u1)θ0" must necessarily be a multiple of the basic contact form θ0 = "du" − "u1dx". Proceeding to the second jet space "J2(π)" with additional coordinate "u2", such that
a general 1-form has the construction
This is a contact form if and only if
which implies that "e" = 0 and "a" = −"bσ′(x)" − "cσ′′(x)". Therefore, θ is a contact form if and only if
where θ1 = "du"1 − "u"2"dx" is the next basic contact form (Note that here we are identifying the form θ0 with its pull-back formula_47 to "J2(π)").
In general, providing "x, u" ∈ R, a contact form on "Jr+1(π)" can be written as a linear combination of the basic contact forms
where
Similar arguments lead to a complete characterization of all contact forms.
In local coordinates, every contact one-form on "Jr+1(π)" can be written as a linear combination
with smooth coefficients formula_51 of the basic contact forms
"|I|" is known as the order of the contact form formula_53. Note that contact forms on "Jr+1(π)" have orders at most "r". Contact forms provide a characterization of those local sections of "πr+1" which are prolongations of sections of π.
Let ψ ∈ Γ"W"("πr+1"), then "ψ" = "jr+1"σ where σ ∈ Γ"W"(π) if and only if formula_54
Vector fields.
A general vector field on the total space "E", coordinated by formula_55, is
A vector field is called horizontal, meaning that all the vertical coefficients vanish, if "formula_57" = 0.
A vector field is called vertical, meaning that all the horizontal coefficients vanish, if "ρi" = 0.
For fixed "(x, u)", we identify
having coordinates "(x, u, ρi, φα)", with an element in the fiber "TxuE" of "TE" over "(x, u)" in "E", called a tangent vector in "TE". A section
is called a vector field on "E" with
and ψ in "Γ(TE)".
The jet bundle "Jr(π)" is coordinated by formula_61. For fixed "(x, u, w)", identify
having coordinates
with an element in the fiber formula_64 of "TJr(π)" over "(x, u, w)" ∈ "Jr(π)", called a tangent vector in "TJr(π)". Here,
are real-valued functions on "Jr(π)". A section
is a vector field on "Jr(π)", and we say formula_67
Partial differential equations.
Let "(E, π, M)" be a fiber bundle. An "r"-th order partial differential equation on π is a closed embedded submanifold "S" of the jet manifold "Jr(π)". A solution is a local section σ ∈ Γ"W"(π) satisfying formula_68, for all "p" in "M".
Consider an example of a first order partial differential equation.
Example.
Let π be the trivial bundle (R2 × R, pr1, R2) with global coordinates ("x"1, "x"2, "u"1). Then the map "F" : "J"1(π) → R defined by
gives rise to the differential equation
which can be written
The particular
has first prolongation given by
and is a solution of this differential equation, because
and so formula_75 for "every" "p" ∈ R2.
Jet prolongation.
A local diffeomorphism "ψ" : "Jr"("π") → "Jr"("π") defines a contact transformation of order "r" if it preserves the contact ideal, meaning that if θ is any contact form on "Jr"("π"), then "ψ*θ" is also a contact form.
The flow generated by a vector field "Vr" on the jet space "Jr(π)" forms a one-parameter group of contact transformations if and only if the Lie derivative formula_76 of any contact form θ preserves the contact ideal.
Let us begin with the first order case. Consider a general vector field "V"1 on "J"1("π"), given by
We now apply formula_78 to the basic contact forms formula_79 and expand the exterior derivative of the functions in terms of their coordinates to obtain:
Therefore, "V1" determines a contact transformation if and only if the coefficients of "dxi" and formula_81 in the formula vanish. The latter requirements imply the contact conditions
The former requirements provide explicit formulae for the coefficients of the first derivative terms in "V1":
where
denotes the zeroth order truncation of the total derivative "Di".
Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if formula_85 satisfies these equations, "Vr" is called the r"-th prolongation of "V" to a vector field on "Jr(π).
These results are best understood when applied to a particular example. Hence, let us examine the following.
Example.
Consider the case "(E, π, M)", where "E" ≅ R2 and "M" ≃ R. Then, "(J1(π), π, E)" defines the first jet bundle, and may be coordinated by "(x, u, u1)", where
for all "p" ∈ "M" and "σ" in Γ"p"("π"). A contact form on "J1(π)" has the form
Consider a vector "V" on "E", having the form
Then, the first prolongation of this vector field to "J1(π)" is
If we now take the Lie derivative of the contact form with respect to this prolonged vector field, formula_90 we obtain
Hence, for preservation of the contact ideal, we require
And so the first prolongation of "V" to a vector field on "J1(π)" is
Let us also calculate the second prolongation of "V" to a vector field on "J2(π)". We have formula_94 as coordinates on "J2(π)". Hence, the prolonged vector has the form
The contact forms are
To preserve the contact ideal, we require
Now, "θ" has no "u"2 dependency. Hence, from this equation we will pick up the formula for "ρ", which will necessarily be the same result as we found for "V1". Therefore, the problem is analogous to prolonging the vector field "V1" to "J"2(π). That is to say, we may generate the "r"-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, "r" times. So, we have
and so
Therefore, the Lie derivative of the second contact form with respect to "V2" is
Hence, for formula_101 to preserve the contact ideal, we require
And so the second prolongation of "V" to a vector field on "J"2(π) is
Note that the first prolongation of "V" can be recovered by omitting the second derivative terms in "V2", or by projecting back to "J1(π)".
Infinite jet spaces.
The inverse limit of the sequence of projections formula_104 gives rise to the infinite jet space "J∞(π)". A point formula_105 is the equivalence class of sections of π that have the same "k"-jet in "p" as σ for all values of "k". The natural projection π∞ maps formula_105 into "p".
Just by thinking in terms of coordinates, "J∞(π)" appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on "J∞(π)", not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections formula_107 of manifolds is the sequence of injections formula_108 of commutative algebras. Let's denote formula_109 simply by formula_110. Take now the direct limit formula_111 of the formula_110's. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object "J∞(π)". Observe that formula_111, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.
Roughly speaking, a concrete element formula_114 will always belong to some formula_110, so it is a smooth function on the finite-dimensional manifold "Jk"(π) in the usual sense.
Infinitely prolonged PDEs.
Given a "k"-th order system of PDEs "E" ⊆ "Jk(π)", the collection "I(E)" of vanishing on "E" smooth functions on "J∞(π)" is an ideal in the algebra formula_110, and hence in the direct limit formula_111 too.
Enhance "I(E)" by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal "I" of formula_111 which is now closed under the operation of taking total derivative. The submanifold "E"(∞) of "J"∞(π) cut out by "I" is called the infinite prolongation of "E".
Geometrically, "E"(∞) is the manifold of formal solutions of "E". A point formula_105 of "E"(∞) can be easily seen to be represented by a section σ whose "k"-jet's graph is tangent to "E" at the point formula_120 with arbitrarily high order of tangency.
Analytically, if "E" is given by φ = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a point "p" that make vanish the Taylor series of formula_121 at the point "p".
Most importantly, the closure properties of "I" imply that "E"(∞) is tangent to the infinite-order contact structure formula_122 on "J∞(π)", so that by restricting formula_122 to "E"(∞) one gets the diffiety formula_124, and can study the associated Vinogradov (C-spectral) sequence.
Remark.
This article has defined jets of local sections of a bundle, but it is possible to define jets of functions "f: M" → "N", where "M" and "N" are manifolds; the jet of "f" then just corresponds to the jet of the section
("grf" is known as the graph of the function "f") of the trivial bundle ("M" × "N", π1, "M"). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π1.