reduction of structure group

reduction of structure group

Given a

fiber bundle

p

:

E

→

B

with typical fiber

F

and structure group

G

(henceforth called an

(

F

,

G

)

-bundle over

B

), we say that the bundle admits a

reduction

of its structure group to

H

,

where

H

<

G

is a

subgroup

, if it is

isomorphic

to an

(

F

,

H

)

-bundle over

B

.

Equivalently,

E

admits a

reduction of structure group

to

H

if there is
a choice of local trivializations covering

E

such that the

transition
functions

all belong to

H

.

Remark 1

Here, the action of

H

on

F

is the

restriction

of the

G

-action; in
particular, this means that an

(

F

,

H

)

-bundle is automatically an

(

F

,

G

)

-bundle. The bundle

isomorphism

in the

definition

then becomes meaningful
in the category of

(

F

,

G

)

-bundles over

B

.

Example 1

Let

H

be the trivial subgroup. Then, the

existence

of a reduction of structure group to

H

is

equivalent

to the bundle being

trivial

.

For the following examples, let

E

be an

n

-dimensional

vector bundle

, so that

F

≅

ℝ

n

with

G

=

G

⁢

L

⁢

(

n

,

ℝ

)

,

the

general linear group

acting as
usual.

Example 2

Set

H

=

G

⁢

L

+

⁢

(

n

,

ℝ

)

,

the subgroup of

G

⁢

L

⁢

(

n

,

ℝ

)

consisting of matrices with

positive

determinant

. A reduction to

H

is equivalent to an

orientation

of the vector bundle. In the case where

B

is a

smooth manifold

and

E

=

T

⁢

B

is its

tangent bundle

, this coincides with other definitions of an orientation of

B

.

Example 3

Set

H

=

O

⁢

(

n

)

, the

orthogonal group

. A reduction to

H

is called a

Riemannian

or

Euclidean structure

on the vector bundle. It coincides with a

continuous

fiberwise choice
of a

positive definite

inner product

, and for the case of the tangent bundle,
with the usual notion of a

Riemannian metric

on a

manifold

.

When

B

is

paracompact

, an

argument

with

partitions of unity

shows that
a Riemannian structure always exists on any given vector bundle. For this reason, it is often convenient to start out assuming the structure group
to be

O

⁢

(

n

)

.

Example 4

Let

n

=

2

⁢

m

be even, and let

H

=

G

⁢

L

⁢

(

m

,

ℂ

)

,

the group of

invertible

complex matrices

, embedded in

G

⁢

L

⁢

(

n

,

ℝ

)

by means of the usual identification of

ℂ

with

ℝ

2

.

A reduction to

H

is called a

complex structure

on the vector bundle, and
it is equivalent to a continuous fiberwise choice of an

endomorphism

J

satisfying

J

2

=

-

I

.

A complex structure on a tangent bundle is called an

almost-complex structure

on the manifold. This is to distinguish it from the
more restrictive notion of a complex structure on a manifold, which requires the existence of an atlas with charts in

ℂ

m

such that the transition functions are holomorphic.

Example 5

Let

H

=

G

⁢

L

⁢

(

1

,

ℝ

)

×

G

⁢

L

⁢

(

n

-

1

,

ℝ

)

,

embedded in

G

⁢

L

⁢

(

n

,

ℝ

)

by

(

A

,

B

)

↦

A

⊕

B

.

A reduction to

H

is equivalent to the
existence of a

splitting

E

≅

E

1

⊕

E

2

,

where

E

1

is a

line bundle

.
More generally, a reduction to

G

⁢

L

⁢

(

k

,

ℝ

)

×

G

⁢

L

⁢

(

n

-

k

,

ℝ

)

is equivalent to a splitting

E

≅

E

1

⊕

E

2

,

where

E

1

is a

k

-plane bundle.

Remark 2

These examples all have two features in common, namely:

•

the subgroup

H

can be interpreted as being precisely the subgroup of

G

which

preserves

a particular

structure

, and,

•

a reduction to

H

is equivalent to a continuous fiber-by-fiber choice of a
structure of the same kind.

For example,

O

⁢

(

n

)

is the subgroup of

G

⁢

L

⁢

(

n

,

ℝ

)

which preserves the
standard inner product of

ℝ

n

,

and reduction of structure to

O

⁢

(

n

)

is
equivalent to a fiberwise choice of inner products.

This is not a coincidence. The intuition behind this is as follows. There
is no obstacle to choosing a fiberwise inner product in a

neighborhood

of any given point

x

∈

B

: we simply choose a neighborhood

U

on which the bundle is trivial, and with respect to a

trivialization

p

-

1

⁢

(

U

)

≅

ℝ

n

×

U

, we can let the inner product on each

p

-

1

⁢

(

y

)

be the standard inner product. However, if we make these choices locally around every point in

B

,
there is no guarantee that they “glue together” properly to yield a global
continuous choice,

unless

the transition functions preserve the standard
inner product. But this is precisely what reduction of structure to

O

⁢

(

n

)

means.

The same explanation holds for subgroups preserving other kinds of structure.

Title

reduction of structure group

Canonical

name

ReductionOfStructureGroup

Date of creation

2013-03-22 13:26:06

Last modified on

2013-03-22 13:26:06

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

12

Author

antonio (1116)

Entry type

Definition

Classification

msc 55R10

Related topic

VectorBundle

Related topic

FiberBundle

Defines

Euclidean structure

Defines

Riemannian structure

Defines

complex structure

Defines

almost-complex structure

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