jet bundle

jet bundle

0.1

Introduction

Let

p

:

X

→

B

be a

surjective

submersion

of

𝒞

r

differential manifolds

, where

r

∈

{

0

,

1

,

…

}

∪

{

∞

,

ω

}

(

𝒞

ω

means

real analytic

). For all integers

k

≥

0

with

k

≤

r

, we will
define a

fibre bundle

J

B

k

⁢

X

over

X

, called the

k

-th

jet
bundle

of

X

over

B

. The fibre of this bundle above a point

x

∈

X

can be interpreted as the set of

equivalence classes

of

local
sections

of

p

x

, where two

sections

are considered

equivalent

if their first

k

derivatives

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 function

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

structure

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. the set of

𝒞

r

functions

s

:

U

→

p

-

1

⁢

U

such that

p

|

p

-

1

⁢

U

∘

s

=

id

U

. Every point of

B

has an open subset

U

such that there
exists at least one section of

p

over

U

, due to the

assumption

that

p

is a surjective submersion. For all

x

∈

X

, we define the
fibre of

J

B

k

⁢

X

above

x

by

J

B

k

⁢

X

⁢

(

x

)

=

{

(

U

,

s

)

:

U

⊂

B

⁢

open

,

x

∈

U

,

s

∈

Γ

⁢

(

U

,

X

)

,

s

⁢

(

p

⁢

(

x

)

)

=

x

}

/

∼

,

where the

equivalence relation

∼

is defined by

s

∼

s

′

⇔

s

and

s

′

induce the same map between the fibres
at

p

⁢

(

x

)

and

x

of the

k

-th iterated

tangent bundles

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,

J

B

k

⁢

X

is defined as the

disjoint
union

of the sets

J

B

k

⁢

X

⁢

(

x

)

with

x

∈

X

. Write

π

:

J

B

k

⁢

X

→

X

for the ‘obvious’ projection map, defined by

π

⁢

(

[

U

,

s

]

)

=

x

⁢

for

⁢

[

U

,

s

]

∈

J

B

k

⁢

X

⁢

(

x

)

.

Notice that

J

B

0

⁢

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

]

∈

J

B

k

⁢

X

⁢

(

s

⁢

(

y

)

)

we obtain a section of

π

∘

p

:

J

B

k

⁢

X

→

B

over

U

for each

k

≥

0

, called the

k

-th

prolongation

of

s

. Composing this section on the left with

π

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

J

B

k

+

1

⁢

X

as the first jet bundle of

J

B

k

⁢

X

over

B

for

k

≥

1

. This is useful because the manifold structure only needs to be
defined for

J

B

1

⁢

X

.

We make

J

B

1

⁢

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

,

ϕ

,

W

′

)

, where

ϕ

is a

diffeomorphism

between open
subsets

W

⊂

X

and

W

′

⊂

ℝ

n

. Without loss of
generality, we assume that

p

⁢

(

W

)

is

contained

in the domain

V

of a
chart

(

V

,

ψ

,

V

′

)

on

B

, with

V

′

⊂

ℝ

m

. Here

n

and

m

are the local

dimensions

of

X

and

B

, respectively.

For all

x

∈

W

and all

[

U

,

s

]

∈

J

B

1

⁢

X

⁢

(

x

)

, we have the

tangent map

T

⁢

s

⁢

(

p

⁢

(

x

)

)

, which is a

linear map

from

T

⁢

V

⁢

(

p

⁢

(

x

)

)

to

T

⁢

W

⁢

(

x

)

.
These

tangent spaces

are

isomorphic

to

ℝ

m

and

ℝ

n

via the chosen charts, so that

T

⁢

s

⁢

(

p

⁢

(

x

)

)

acts as a
matrix

M

x

⁢

(

[

U

,

s

]

)

∈

ℝ

n

×

m

:

\xymatrix

⁢

T

⁢

W

⁢

(

x

)

⁢

\ar

⁢

[

r

]

≃

⁢

&

⁢

ℝ

n

⁢

T

⁢

V

⁢

(

p

⁢

(

x

)

)

⁢

\ar

⁢

[

u

]

T

⁢

s

⁢

(

p

⁢

(

x

)

)

⁢

\ar

⁢

[

r

]

≃

⁢

&

⁢

ℝ

m

⁢

\ar

⁢

[

u

]

M

x

⁢

(

[

U

,

s

]

)

The definition of the equivalence relation

∼

on

J

B

1

⁢

X

⁢

(

x

)

means
that the association

[

U

,

s

]

↦

M

x

⁢

(

[

U

,

s

]

)

is well-defined and

injective

for each

x

∈

W

. The image of

M

x

consists of the
matrices with the property that multiplying them on the left with the

m

×

n

matrix corresponding to the tangent map

T

⁢

p

⁢

(

x

)

gives
the

m

×

m

identity matrix

. These matrices form a

m

⁢

(

n

-

m

)

-dimensional

linear subspace

L

x

⊂

ℝ

n

×

m

,
and

L

=

⋃

x

∈

W

{

x

}

×

L

x

is a

submanifold

of

W

×

ℝ

n

×

m

.

We both the differentiable structure of

π

-

1

⁢

U

and a local trivialisation of

J

B

1

⁢

X

as a

vector bundle

by requiring that

π

-

1

⁢

W

→

L

[

U

,

s

]

↦

(

π

⁢

(

[

U

,

s

]

)

,

M

π

⁢

(

[

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

M

x

⁢

(

[

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

J

B

1

⁢

X

.

Iterating the above construction by defining

J

B

k

+

1

⁢

X

as the first
jet bundle of

J

B

k

⁢

X

over

B

, each jet bundle

J

B

k

+

1

⁢

X

becomes
a vector bundle over

J

B

k

⁢

X

and a fibre bundle over

X

. Normally,
only

J

B

1

⁢

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

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