Connection Form

PRINCIPAL BUNDLES

For a principal ''G''-bundle

E	o B

, for each

x\in E

let

T_x(E)

denote the tangent space at ''x'' and

V_x

the ''vertical'' subspace tangent to the fiber . Then connection is an assignment of a ''horizontal'' subspace

H_x

T_x(E)

such that

#

T_x(E)

is direct sum of

V_x

and

H_x

,
# The distribution of

H_x

is invariant with respect to the ''G''-action on ''E'', i.e.

H_{ax}=D_x(R_a)H_{x}

for any

x\in E

and

a\in G

, here

D_x(R_a)

denotes the differential of the Group Action by ''a'' at ''x''.
#The distribution

H_x

depends smoothly on ''x''.

This can be recast more elegantly using the Jet Bundle

JE 
ightarrow E

. The assignment of a horizontal subspace at each point is none other than a smooth section of this jet bundle.

The one-parameter subgroups of ''G'' act vertically on ''E''. The differential of this action allows one to identify the subspace

V_x

with the Lie algebra ''g'' of group ''G'', say by map

\iota:V_x	o g

.
Then the connection form is a form

\omega

E

with values in ''g'' defined by

\omega(X)=\iota\circ v(X)

where

v

denotes projection at

x \in E

X \in T_x

V_x

with kernel

H_x

.

The connection form satisfies the following two properties:

The connection transforms Equivariantly under the ''G'' action: $R_h^---\omega=\hbox{Ad}(h^{-1})\omega$ for all ''h''∈''G''.

The connection maps vertical vector fields to their associated elements of the Lie algebra: $\omega(X)=\iota(X)$ for all ''X''∈''V''.

Given a local trivialization one can reduce

\omega

to the horizontal vector fields (in this trivialization). It defines form say

\omega'

on ''B'' via Pullback . The form

\omega'

defines

\omega

completely, but it depends on the choice of trivialization. (This form is often also called a connection form and denoted also by

\omega.

)

Related definitions

Exterior covariant derivative

The Exterior Covariant Derivative is a very useful notion which makes it possible to simplify formulas using a connection.
Given a tensor-valued differential ''k''-form

\phi

its exterior covariant derivative

D\phi

is defined by
:

D\phi(X_0,X_1,...,X_k) := d\phi(h(X_0),h(X_1),...,h(X_k)),

where ''h'' denotes the projection to the horizontal subspace,

H_x

with kernel

V_x

and

X_i

are arbitrary vector fields on ''E''.

Curvature form

The Curvature Form

\Omega

, a ''g''-valued 2-form, can be defined by

:

\Omega=d\omega +{1\over 2}

{Link without Title} =D\omega,

,---" class="copylinks" target="_blank">{Link without Title} denotes the Lie Bracket . This equation is also called the ''second structure equation''.

Torsion

For the connection on a Frame Bundle ,
the curvature is not the only invariant of connection since the additional structure should be taken into account. Namely one has an extra canonical Rⁿ-valued form

heta=	heta^i

on ''E'' defined by identity

:

X=\sum_i	heta^i(X)e_i.\,

Then the Torsion Form , an Rⁿ-valued 2-form can be defined by

:

\Theta=d	heta+{1\over 2}[\omega, 	heta]=D	heta.

This equation is also called the ''first structure equation''.

VECTOR BUNDLES

The connection form for the vector bundle is the form on the total space of the Associated principal bundle, but it can also
be completely described by the following form (on the base in a not invariant way). This subsection can be considered as a smoother but somewhat inaccurate introduction to connection forms.

A Covariant Derivative on a Vector Bundle is a way to "differentiate" bundle sections along tangent vectors; it is also sometimes called a Connection . Let

\zeta:E	o B

be a vector bundle over a smooth manifold

B

with an ''n''-dimensional vector space

F

as a fiber. Let us denote by

abla_uv

a section of the vector bundle, the result of differentiation of the section of vector bundle

v

along the tangent vector field

u

. In order to be a covariant derivative,

abla

must satisfy the following identities:

:(i)

abla_u(v_1+v_2)=
abla_uv_1+
abla_uv_2

and

abla_{u_1+u_2}v=
abla_{u_1}v+
abla_{u_2}v

(linearity)

:(ii)

abla_u(fv)=df(u) v +f
abla_uv

and

abla_{f u}v=f
abla_{u}v

for any smooth function

f.

The simplest example: if

\zeta:E=F	imes B 	o B

is the projection, i.e.

\zeta

is a trivial vector bundle, then
any section can be described by a smooth map

v:B	o F

. Therefore, one can consider the trivial covariant derivative defined by partial derivatives:

abla_u v=\partial v/\partial u.

If one has two connections

abla

and

abla'

on the same vector bundle then the difference

\omega(u)v=
abla_uv-
abla'_uv

depends only on values of ''u'' and ''v'' at a point.

\omega

is a 1- Form on

B

with values in

Hom(F,F)

;
i.e.

\omega(u)\in Hom(F,F)

and

\omega

can be described as an

n	imes n

-matrix of one-forms.
In particular, if one chooses a local trivialization of the vector bundle and takes

abla'

to be the corresponding trivial connection, then

\omega

gives a complete local description of

abla

.

The choice of trivialization is equivalent to choosing frames in each fiber; this explains the reason for the name '' Method Of Moving Frames ''. Let us choose (a local smooth section of) basis frames

e_i

in fibers. Then the matrix of 1-forms

\omega=\omega_i^j

is defined by the following identity:

:

abla_u e_i=\sum_j\omega^j_i(u)e_j.

G\subset GL(F)

is the Structure Group of the vector bundle and the connection

abla

respects the group structure then the form

\omega

is a 1-form with values in

g

, the Lie Algebra of

G

.
In particular, for the Tangent Bundle of a Riemannian Manifold we have

O(n)

as the structure group and the form

\omega

for the Levi-Civita Connection takes values in ''so''(''n''), the Lie algebra of

O(n)

(which can be thought of as antisymmetric matrices in an orthonormal basis).

Related definitions

Curvature

The connection form (

\omega

) describes a connection (

abla

) in a non-invariant way; it depends on the choice of local trivialization.
The following construction extracts invariant information out of

\omega

:

A 2-form with values in

Hom(F,F)

is called Curvature Form if it can be written as

:

\Omega=d\omega +\omega\wedge\omega,

where

d

stands for Exterior Derivative and

\wedge

is the Wedge Product . This equation also called the ''second structure equation''.

Torsion

For the connection on tangent bundle, the curvature is not the only invariant of the connection since the additional structure should be taken into account. Namely, one has an extra canonical Rⁿ-valued form

heta=	heta^i

on ''B'' defined by identity

:

X=\sum_i	heta^i(X)e_i.

Then the torsion, an Rⁿ-valued 2-form, can be defined by

:

\Theta=d	heta+\omega\wedge 	heta\ \ \mbox{or} \ \ \Theta^i=d	heta^i+\sum_j\omega^i_j\wedge 	heta^j.

This equation is also called the ''first structure equation''.

SEE ALSO

Cartan Connection

REFERENCES

Kobayashi, Shoshichi; Nomizu, Katsumi; ''Foundations of differential geometry'' Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xii+329 pp. ISBN 0-471-15733-3

	CATEGORIES ABOUT CONNECTION FORM

	differential geometry

	fiber bundles

	connection mathematics

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Connection Form