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 newlyintroduced 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.)
More recently, 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.
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Let be a fiber bundle in a category of manifolds and let , with . Let denote the set of all local sections whose domain contains . Let be a multiindex (an ordered mtuple of integers), then
Define the local sections to have the same jet at if
The relation that two maps have the same rjet is an equivalence relation. An rjet is an equivalence class under this relation, and the rjet with representative is denoted . The integer r is also called the order of the jet.
is the source of .
is the target of .
The jet manifold of is the set
and is denoted . We may define projections and called the source and target projections respectively, by
If , then the kjet projection is the function defined by
From this definition, it is clear that and that if , then . It is conventional to regard , the identity map on and to identify with .
The functions and are smooth surjective submersions.
A coordinate system on will generate a coordinate system on . Let be an adapted coordinate chart on , where . The induced coordinate chart on is defined by
where
and the functions
are specified by
and are known as the derivative coordinates.
Given an atlas of adapted charts on , the corresponding collection of charts is a finitedimensional atlas on .
Since the atlas on each defines a manifold, the triples and all define fibered manifolds. In particular, if is a fiber bundle, the triple defines the jet bundle of .
If is an open submanifold, then
If , then the fiber is denoted .
Let be a local section of with domain . The jet prolongation of is the map defined by
Note that , so really is a section. In local coordinates, is given by
We identify with .
If is the trivial bundle , then there is a canonical diffeomorphism between the first jet bundle and . To construct this diffeomorphism, for each write .
Then, whenever
Consequently, the mapping
is welldefined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if are coordinates on , where is the identity coordinate, then the derivative coordinates on correspond to the coordinates on .
Likewise, if is the trivial bundle , then there exists a canonical diffeomorphism between and
A differential 1form on the space is called a contact form (ie. ) if it is pulled back to the zero form on by all prolongations. In other words, if , then if and only if, for every open submanifold and every
The distribution on generated by the contact forms is called the Cartan distribution. It 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 not involutive and are of growing dimension when passing to higher order jet spaces. Surprisingly though, when passing to the space of infinite order jets this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold .
Let us consider the case , where and . Then, defines the first jet bundle, and may be coordinated by , where
for all and . A general 1form on takes the form
A section has first prolongation . Hence, can be calculated as
This will vanish for all sections if and only if and . Hence, must necessarily be a multiple of the basic contact form . Proceeding to the second jet space with additional coordinate , such that
a general 1form has the construction
This is a contact form if and only if
which implies that and . Therefore, is a contact form if and only if
where is the next basic contact form (Note that here we are identifying the form with its pullback to ).
In general, providing , a contact form on 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 oneform on can be written as a linear combination
with smooth coefficients of the basic contact forms
is known as the order of the contact form . Note that contact forms on have orders at most . Contact forms provide a characterization of those local sections of which are prolongations of sections of .
Let , then where if and only if
A general vector field on the total space , coordinated by , is
A vector field is called horizontal, meaning all the vertical coefficients vanish, if .
A vector field is called vertical, meaning all the horizontal coefficients vanish, if .
For fixed , we identify
having coordinates , with an element in the fiber of over , called a tangent vector in . A section
is called a vector field on with and .
The jet bundle is coordinated by . For fixed , identify
having coordinates , with an element in the fiber of over , called a tangent vector in . Here, are realvalued functions on . A section
is a vector field on , and we say .
Let be a fiber bundle. An order partial differential equation on is a closed embedded submanifold of the jet manifold . A solution is a local section satisfying .
Let us consider an example of a first order partial differential equation.
Let be the trivial bundle with global coordinates . Then the map defined by
gives rise to the differential equation
which can be written
The particular section defined by
has first prolongation given by
and is a solution of this differential equation, because
and so for every .
A local diffeomorphism defines a contact transformation of order if it preserves the contact ideal, meaning that if is any contact form on , then is also a contact form.
The flow generated by a vector field on the jet space forms a oneparameter group of contact transformations if and only if the Lie derivative of any contact form preserves the contact ideal.
Let us begin with the first order case. Consider a general vector field on , given by
We now apply to the basic contact forms , and obtain
where we have expanded the exterior derivative of the functions in terms of their coordinates. Next, we note that
and so we may write
Therefore, determines a contact transformation if and only if the coefficients of and 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 :
denotes the zeroth order truncation of the total derivative .
Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if satisfies these equations, is called the prolongation of to a vector field on .
These results are best understood when applied to a particular example. Hence, let us examine the following.
Let us consider the case , where and . Then, defines the first jet bundle, and may be coordinated by , where
for all and . A contact form on has the form
Let us consider a vector on , having the form
Then, the first prolongation of this vector field to is
If we now take the Lie derivative of the contact form with respect to this prolonged vector field, , we obtain
But, we may identify . Thus, we get
Hence, for to preserve the contact ideal, we require
And so the first prolongation of to a vector field on is
Let us also calculate the second prolongation of to a vector field on . We have as coordinates on . Hence, the prolonged vector has the form
The contacts forms are
To preserve the contact ideal, we require
Now, has no dependency. Hence, from this equation we will pick up the formula for , which will necessarily be the same result as we found for . Therefore, the problem is analogous to prolonging the vector field to . That is to say, we may generate the prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, times. So, we have
and so
Therefore, the Lie derivative of the second contact form with respect to is
Again, let us identify and . Then we have
Hence, for to preserve the contact ideal, we require
And so the second prolongation of to a vector field on is
Note that the first prolongation of can be recovered by omitting the second derivative terms in , or by projecting back to .
The inverse limit of the sequence of projections gives rise to the infinite jet space . A point is the equivalence class of sections of π that have the same kjet in p as σ for all values of k. The natural projection maps into p.
Just by thinking in terms of coordinates, appears to be an infinitedimensional geometric object. In fact, the simplest way of introducing a differentiable structure on , not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections of manifolds is the sequence of injections of commutative algebras. Let's denote simply by . Take now the direct limit of the 's. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object . Observe that , being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.
Roughly speaking, a concrete element will always belong to some , so it is a smooth function on the finitedimensional manifold J^{k}(π) in the usual sense.
Given a kth order system of PDE's , the collection of vanishing on smooth functions on is an ideal in the algebra , and hence in the direct limit too.
Enhance by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal I of which is now closed under the operation of taking total derivative. The submanifold of cut out by I is called the infinite prolongation of .
Geometrically, is the manifold of formal solutions of . A point of can be easily seen to be represented by a section σ whose kjet's graph is tangent to at the point with arbitrarily high order of tangency.
Analytically, if is given by , 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 at the point p.
Most importantly, the closure properties of I imply that is tangent to the infiniteorder contact structure on , so that by restricting to one gets the diffiety , and can study the associated Cspectral sequence.
This article has defined jets of local sections of a bundle, but it is possible to define jets of functions , where and are manifolds; the jet of then just corresponds to the jet of the section
( is known as the graph of the function ) of the trivial bundle . However, this restriction does not simplify the theory, as the global triviality of does not imply the global triviality of .
