
Jet bundle
Encyclopedia
In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
out of a given smooth fiber bundle. It makes it possible to write differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s 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 Ehresmann, and were an advance on the method (prolongation) of Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
, of dealing geometrically with higher derivatives
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, by imposing differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
conditions on newly-introduced formal variables. Jet bundles are sometimes called sprays, although sprays
Spray (mathematics)
In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→ΦHt∈TM obey the rule...
usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifold
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold together with the structure of an intrinsic quasimetric space in which the length of any rectifiable curve is given by the length functional...
s.)
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
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory
Covariant classical field theory
In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented by sections of fiber bundles and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the...
and much work is done in general relativistic
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
formulations of fields using this approach.
Jets
Let
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
in a category of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s and let


Let






Define the local sections




The relation that two maps have the same

Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. An r-jet is an equivalence class under this relation, and the r-jet with representative







Jet manifolds
The


and is denoted



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If



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From this definition, it is clear that




Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
on



The functions


Smooth
Smooth means having a texture that lacks friction. Not rough.Smooth may also refer to:-In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
surjective submersion
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
s.
A coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
on







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where
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and the


are specified by

and are known as the derivative coordinates.
Given an atlas of adapted charts





Jet bundles
Since the atlas on each


In particular, if




If


If



Let







Note that




We identify


Example
If

Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
between the first jet bundle


To construct this diffeomorphism, for each


Then, whenever


Consequently, the mapping
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is well-defined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if







Likewise, if




Contact forms
A differential 1-form



In other words, if


If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
, for every open submanifold



The distribution
Distribution (differential geometry)
In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties...
on

Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s. The Cartan distributions are not involutive
Distribution (differential geometry)
In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties...
and are of growing dimension when passing to higher order jet spaces. Surprisingly though, when passing to the space of infinite order jets


Example
Let us consider the case


Then,


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for all




A section


Hence,

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This will vanish for all sections





Proceeding to the second jet space



a general 1-form has the construction

This is a contact form if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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which implies that




where

(Note that here we are identifying the form



In general, providing


Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
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


with smooth coefficients






Contact forms provide a characterization of those local sections of


Let




Vector fields
A general vector fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
on the total space



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


having coordinates




Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
in

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is called a vector field on



The jet bundle



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having coordinates





Here,


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is a vector field on


Partial differential equations
Let

Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
on

Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
submanifold


A solution is a local section


Let us consider an example of a first order partial differential equation.
Example
Let


Then the map


gives rise to the differential equation

which can be written

The particular section


has first prolongation given by

and is a solution of this differential equation, because
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and so


Jet Prolongation
A local diffeomorphism




The flow generated by a vector field


Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...


Let us begin with the first order case. Consider a general vector field



We now apply


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where we have expanded the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
of the functions in terms of their coordinates.
Next, we note that

and so we may write
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Therefore,



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





These results are best understood when applied to a particular example. Hence, let us examine the following.
Example
Let us consider the case


Then,


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for all




Let us consider a vector



Then, the first prolongation of this vector field to

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If we now take the Lie derivative of the contact form with respect to this prolonged vector field,

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But, we may identify

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Hence, for

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And so the first prolongation of



Let us also calculate the second prolongation of


We have



The contacts forms are
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To preserve the contact ideal, we require
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Now,






That is to say, we may generate the


So, we have

and so
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Therefore, the Lie derivative of the second contact form with respect to

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Again, let us identify


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Hence, for

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And so the second prolongation of



Note that the first prolongation of



Infinite Jet Spaces
The inverse limitInverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
of the sequence of projections











Just by thinking in terms of coordinates,


Differential calculus over commutative algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms...
. Dual to the sequence of projections


of commutative algebras. Let's denote


Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...




Roughly speaking, a concrete element



Infinitely prolonged PDE's
Given a




Ideal
-In philosophy:* Ideal , values that one actively pursues as goals* Platonic ideal, a philosophical idea of trueness of form, associated with Plato-In mathematics:* Ideal , special subsets of a ring considered in abstract algebra...
in the algebra


Enhance

Total derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...
s applied to all its elements. This way we get a new ideal






Geometrically,








Analytically, if




Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
of


Most importantly, the closure properties of






Diffiety
In mathematics a diffiety, is a geometrical object introduced by playing the same role in the modern theory of partial differential equations as algebraic varieties play for algebraic equations....

Remark
This article has defined jets of local sections of a bundle, but it is possible to define jets of functions



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(




