NationMaster - Encyclopedia: Jet bundle

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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

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. 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.) Charles Ehresmann (1905-1979) was a French mathematician who worked on differential topology and category theory. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space cares natural metric then geodesics are defined to be (locally) the shortest path between points on manifolds. ... In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent...

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. Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... In recent years, there has been renewed interest in the covariant formalism of classical field theory. ... It has been suggested that Einsteins theory of gravitation be merged into this article or section. ...

1 Jets
2 Jet manifolds
3 Jet bundles
- 3.1 Example
4 Contact forms
- 4.1 Example
5 Vector fields
6 Partial differential equations
- 6.1 Example
7 Jet Prolongation
- 7.1 Example
8 Remark
9 References

Jets

Main article: Jet (mathematics).

Let $(mathcal{E}, pi, mathcal{M})$ be a fiber bundle in a category of manifolds and let , with $dimmathcal{M}=m$ . Let $Gamma(pi),$ denote the set of all local sections whose domain contains $p,$ . Let $I=(I(1),I(2),ldots,I(m))$ be a multi-index (an ordered $m$ -tuple of integers), then In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

$|I| := sum_{i=1}^{m} I(i)$

$frac{partial^{|I|}}{partial x^{I}} := prod_{i=1}^{m} left( frac{partial}{partial x^{i}} right)^{I(i)}.$

Define the local sections $sigma, eta in Gamma(pi)$ to have the same -jet at $p,$ if

$left.frac{partial^{|I|} sigma^{alpha}}{partial x^{I}}right|_{p} = left.frac{partial^{|I|} eta^{alpha}}{partial x^{I}}right|_{p}, quad 1 leq |I| leq r$

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 is denoted $j^{r}_{p}sigma$ . In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

$p,$ is the source of $j^{r}_{p}sigma$ .

$sigma,(p)$ is the target of $j^{r}_{p}sigma$ .

Jet manifolds

The jet manifold of is the set

${j^{r}_{p}sigma:p in mathcal{M}, sigma in Gamma(pi)}$

and is denoted . We may define projections $pi_{r},$ and $pi_{r,0},$ called the source and target projections respectively, by

$pi_{r}:J^{r}pi,$	$longrightarrow mathcal{M}$
$j^{r}_{p}sigma$	$longmapsto p$


$pi_{r,0}:J^{r}pi,$	$longrightarrow mathcal{E}$
$j^{r}_{p}sigma$	$longmapsto sigma(p)$

If $1 leq k leq r$ , then the $k$ -jet projection is the function $pi_{r,k},$ defined by

$pi_{r,k}:J^{r}pi ,$	$longrightarrow J^{k}pi$
$j^{r}_{p}sigma$	$longmapsto j^{k}_{p}sigma$

From this definition, it is clear that $pi_{r} = pi circ pi_{r,0}$ and that if $0 leq m leq k$ , then $pi_{r,m} = pi_{k,m} circ pi_{r,k}$ . It is conventional to regard $pi_{r,r}=id_{J^{r}pi},$ , the identity map on $J^{r}pi ,$ and to identify $J^{0}pi,$ with . An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

The functions $pi_{r,k}, pi_{r,0},$ and $pi_{r},$ are smooth surjective submersions. Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, a differentiable map f from an m-manifold M to an n-manifold N is called a submersion if its differential df is a surjective map at every point p of M, or equivalently if rank df(p) = dim N. Examples include the projections in smooth vector bundles...

A co-ordinate system on will generate a co-ordinate system on . Let $(U,u),$ be an adapted co-ordinate chart on , where $u = (x^{i}, u^{alpha}),$ . The induced co-ordinate chart $(U^{r}, u^{r}),$ on is defined by Image File history File links Bundle. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

$U^{r} ,$	$= { j^{r}_{p}sigma: sigma(p) in U } ,$
$u^{r} ,$	$= (x^{i}, u^{alpha}, u^{alpha}_{I}),$

where

$x^{i}(j^{r}_{p}sigma) ,$	$= x^{i}(p) ,$
$u^{alpha}(j^{r}_{p}sigma) ,$	$= u^{alpha}(sigma(p)) ,$

and the $n left( {}^{m+r}C_{r} -1right),$ functions

$u^{alpha}_{I}:U^{k} longrightarrow mathbb{R},$

are specified by

$u^{alpha}_{I}(j^{r}_{p}sigma) = left.frac{partial^{|I|} sigma^{alpha}}{partial x^{|I|}}right|_{p}$

and are known as the derivative co-ordinates.

Given an atlas of adapted charts $(U,u),$ on , the corresponding collection of charts $(U^{r},u^{r}),$ is a finite-dimensional $C^{infty},$ atlas on . In mathematics, the dimension of a vector space V is the cardinality (i. ...

Jet bundles

Since the atlas on each defines a manifold, the triples $(J^{r}pi, pi_{r,k}, J^{k}pi), (J^{r}pi, pi_{r,0}, mathcal{E}),$ and $(J^{r}pi, pi_{r}, mathcal{M}),$ all define fibered manifolds. In particular, if $(mathcal{E}, pi, mathcal{M}),$ is a fiber bundle, the triple $(J^{r}pi, pi_{r}, mathcal{M}),$ defines the jet bundle of .

If is an open submanifold, then

$J^{r}left(pi|_{pi^{-1}(W)}right) cong pi^{-1}_{r}(W),$

If $p in mathcal{M},$ , then the fiber $pi^{-1}_{r}(p),$ is denoted $J^{r}_{p}pi,$ .

Let be a local section of with domain . The jet prolongation of is the map $j^{r}sigma:W longrightarrow J^{r}pi,$ defined by

$(j^{r}sigma)(p) = j^{r}_{p}sigma ,$

Note that $pi_{r} circ j^{r}sigma = id_{W} ,$ , so $j^{r}sigma,$ really is a section. In local co-ordinates, is given by

$left(sigma^{alpha}, frac{partial^{|I|} sigma^{alpha}}{partial x^{|I|}}right) qquad 1 leq |I| leq r ,$

We identify $j^{0}sigma,$ with .

Example

If is the trivial bundle $(mathcal{M} times mathbb{R}, pr_{1}, mathcal{M})$ , then there is a canonical diffeomorphism between the first jet bundle and $T^{*}mathcal{M} times mathbb{R}$ . To construct this diffeomorphism, for each write . In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Then, whenever

Consequently, the mapping

is well-defined and is clearly injective. Writing it out in co-ordinates shows that it is a diffeomorphism, because if are co-ordinates on , where is the identity co-ordinate, then the derivative co-ordinates on correspond to the co-ordinates on . In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

Likewise, if is the trivial bundle , then there exists a canonical diffeomorphism between and

Contact forms

A differential 1-form 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 iff, for every open submanifold and every , A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. ... This article discusses the pullback in differential geometry. ... â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

Example

Let us consider the case , where and . Then, defines the first jet bundle, and may be co-ordinated by , where

for all and . A general 1-form on takes the form

A section has first prolongation . Hence, can be calculated as

This will vanish for all sections iff and . Hence, must necessarily be a multiple of the basic contact form . Proceeding to the second jet space with additional co-ordinate , such that

a general 1-form has the construction

This is a contact form iff â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

which implies that and . Therefore, is a contact form iff

where is the next basic contact form (Note that here we are identifying the form with its pull-back to ).

In general, providing , a contact form on can be written as a linear combination of the basic contact forms In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

where .

Similar arguments lead to a complete characterization of all contact forms.

In local coordinates, every contact one-form 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 iff

Vector fields

A general vector field on the total space , co-ordinated by , is Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean 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 , we identify

having co-ordinates , with an element in the fiber of over , called a tangent vector in . A section In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

is called a vector field on with and .

The jet bundle is co-ordinated by . For fixed , identify

having co-ordinates , with an element in the fiber of over , called a tangent vector in . Here, are real-valued functions on . A section

is a vector field on , and we say .

Partial differential equations

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 . In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, see embedding. ...

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

Example

Let be the trivial bundle with global co-ordinates . 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 .

Jet Prolongation

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 one-parameter group of contact transformations iff the Lie derivative of any contact form preserves the contact ideal. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

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 co-ordinates. Next, we note that In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

and so we may write

Therefore, determines a contact transformation iff 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 :

where

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.

Example

Let us consider the case , where and . Then, defines the first jet bundle, and may be co-ordinated 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 co-ordinates 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 .

Remark

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 .

References

Ehresmann, C., "Introduction a la théorie des structures infinitésimales et des pseudo-groupes de mathcal{L}." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.
Kolár(, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4.
Saunders, D.J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
Bocharov, A.V. [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958
Olver, P.J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1

Categories: Differential topology | Differential equations | Fiber bundles

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Contents

Jets GA_googleFillSlot("encyclopedia_square");

Jet manifolds

Jet bundles

Example

Contact forms

Example

Vector fields

Partial differential equations

Example

Jet Prolongation

Example

Remark

References

Jets