Natural bundle

In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle

F

r

(

M

)

{\displaystyle F^{r}(M)}

, for some

r

≥

1

{\displaystyle r\geq 1}

. In other words, its transition functions depend functionally on local changes of coordinates in the base manifold

M

{\displaystyle M}

together with their partial derivatives up to order at most

r

{\displaystyle r}

.

The concept of a natural bundle was introduced in 1972 by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.

Definition

Let

M

f

{\displaystyle {\mathcal {M}}f}

denote the category of smooth manifolds and smooth maps and

M

f

n

{\displaystyle {\mathcal {M}}f_{n}}

the category of smooth

n

{\displaystyle n}

-dimensional manifolds and local diffeomorphisms. Consider also the category

F

M

{\displaystyle {\mathcal {FM}}}

of fibred manifolds and bundle morphisms, and the functor

B

:

F

M

→

M

f

{\displaystyle B:{\mathcal {FM}}\to {\mathcal {M}}f}

associating to any fibred manifold its base manifold.

A natural bundle (or bundle functor) is a functor

F

:

M

f

n

→

F

M

{\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}}

satisfying the following three properties:

B

∘

F

=

i

d

{\displaystyle B\circ F=\mathrm {id} }

, i.e.

F

(

M

)

{\displaystyle F(M)}

is a fibred manifold over

M

{\displaystyle M}

, with projection denoted by

p

M

:

F

(

M

)

→

M

{\displaystyle p_{M}:F(M)\to M}

;

if

U

⊆

M

{\displaystyle U\subseteq M}

is an open submanifold, with inclusion map

i

:

U

↪

M

{\displaystyle i:U\hookrightarrow M}

, then

F

(

U

)

{\displaystyle F(U)}

coincides with

p

M

−

1

(

U

)

⊆

F

(

M

)

{\displaystyle p_{M}^{-1}(U)\subseteq F(M)}

, and

F

(

i

)

:

F

(

U

)

→

F

(

M

)

{\displaystyle F(i):F(U)\to F(M)}

is the inclusion

p

−

1

(

U

)

↪

F

(

M

)

{\displaystyle p^{-1}(U)\hookrightarrow F(M)}

;

for any smooth map

f

:

P

×

M

→

N

{\displaystyle f:P\times M\to N}

such that

f

(

p

,

⋅

)

:

M

→

N

{\displaystyle f(p,\cdot ):M\to N}

is a local diffeomorphism for every

p

∈

P

{\displaystyle p\in P}

, then the function

P

×

F

(

M

)

→

F

(

N

)

,

(

p

,

x

)

↦

F

(

f

(

p

,

⋅

)

)

(

x

)

{\displaystyle P\times F(M)\to F(N),(p,x)\mapsto F(f(p,\cdot ))(x)}

is smooth.

As a consequence of the first condition, one has a natural transformation

p

:

F

→

i

d

M

f

n

{\displaystyle p:F\to \mathrm {id} _{{\mathcal {M}}f_{n}}}

.

Finite order natural bundles

A natural bundle

F

:

M

f

n

→

F

M

{\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}}

is called of finite order

r

{\displaystyle r}

if, for every local diffeomorphism

f

:

M

→

N

{\displaystyle f:M\to N}

and every point

x

∈

M

{\displaystyle x\in M}

, the map

F

(

f

)

x

:

F

(

M

)

x

→

F

(

N

)

f

(

x

)

{\displaystyle F(f)_{x}:F(M)_{x}\to F(N)_{f(x)}}

depends only on the jet

j

x

r

f

{\displaystyle j_{x}^{r}f}

. Equivalently, for every local diffeomorphisms

f

,

g

:

M

→

N

{\displaystyle f,g:M\to N}

and every point

x

∈

M

{\displaystyle x\in M}

, one has

j

x

r

f

=

j

x

r

g

⇒

F

(

f

)

|

F

(

M

)

x

=

F

(

g

)

|

F

(

M

)

x

.