Operation on fibered manifolds
In differential geometry , a fibered manifold is surjective submersion of smooth manifolds Y → X . Locally trivial fibered manifolds are fiber bundles . Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.
Formal definition [ edit ]
Let π ぱい : Y → X be a fibered manifold. A generalized connection on Y is a section Γ がんま : Y → J1 Y , where J1 Y is the jet manifold of Y .[1]
Connection as a horizontal splitting [ edit ]
With the above manifold π ぱい there is the following canonical short exact sequence of vector bundles over Y :
0
→
V
Y
→
T
Y
→
Y
×
X
T
X
→
0
,
{\displaystyle 0\to \mathrm {V} Y\to \mathrm {T} Y\to Y\times _{X}\mathrm {T} X\to 0\,,}
(1 )
where TY and TX are the tangent bundles of Y , respectively, VY is the vertical tangent bundle of Y , and Y ×X TX is the pullback bundle of TX onto Y .
A connection on a fibered manifold Y → X is defined as a linear bundle morphism
Γ がんま
:
Y
×
X
T
X
→
T
Y
{\displaystyle \Gamma :Y\times _{X}\mathrm {T} X\to \mathrm {T} Y}
(2 )
over Y which splits the exact sequence 1 . A connection always exists.
Sometimes, this connection Γ がんま is called the Ehresmann connection because it yields the horizontal distribution
H
Y
=
Γ がんま
(
Y
×
X
T
X
)
⊂
T
Y
{\displaystyle \mathrm {H} Y=\Gamma \left(Y\times _{X}\mathrm {T} X\right)\subset \mathrm {T} Y}
of TY and its horizontal decomposition TY = VY ⊕ HY .
At the same time, by an Ehresmann connection also is meant the following construction. Any connection Γ がんま on a fibered manifold Y → X yields a horizontal lift Γ がんま ∘ τ たう of a vector field τ たう on X onto Y , but need not defines the similar lift of a path in X into Y . Let
R
⊃
[
,
]
∋
t
→
x
(
t
)
∈
X
R
∋
t
→
y
(
t
)
∈
Y
{\displaystyle {\begin{aligned}\mathbb {R} \supset [,]\ni t&\to x(t)\in X\\\mathbb {R} \ni t&\to y(t)\in Y\end{aligned}}}
be two smooth paths in X and Y , respectively. Then t → y (t ) is called the horizontal lift of x (t ) if
π ぱい
(
y
(
t
)
)
=
x
(
t
)
,
y
˙
(
t
)
∈
H
Y
,
t
∈
R
.
{\displaystyle \pi (y(t))=x(t)\,,\qquad {\dot {y}}(t)\in \mathrm {H} Y\,,\qquad t\in \mathbb {R} \,.}
A connection Γ がんま is said to be the Ehresmann connection if, for each path x ([0,1]) in X , there exists its horizontal lift through any point y ∈ π ぱい −1 (x ([0,1])) . A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.
Connection as a tangent-valued form [ edit ]
Given a fibered manifold Y → X , let it be endowed with an atlas of fibered coordinates (xμ みゅー , yi ) , and let Γ がんま be a connection on Y → X . It yields uniquely the horizontal tangent-valued one-form
Γ がんま
=
d
x
μ みゅー
⊗
(
∂
μ みゅー
+
Γ がんま
μ みゅー
i
(
x
ν にゅー
,
y
j
)
∂
i
)
{\displaystyle \Gamma =dx^{\mu }\otimes \left(\partial _{\mu }+\Gamma _{\mu }^{i}\left(x^{\nu },y^{j}\right)\partial _{i}\right)}
(3 )
on Y which projects onto the canonical tangent-valued form (tautological one-form or solder form )
θ しーた
X
=
d
x
μ みゅー
⊗
∂
μ みゅー
{\displaystyle \theta _{X}=dx^{\mu }\otimes \partial _{\mu }}
on X , and vice versa . With this form, the horizontal splitting 2 reads
Γ がんま
:
∂
μ みゅー
→
∂
μ みゅー
⌋
Γ がんま
=
∂
μ みゅー
+
Γ がんま
μ みゅー
i
∂
i
.
{\displaystyle \Gamma :\partial _{\mu }\to \partial _{\mu }\rfloor \Gamma =\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\,.}
In particular, the connection Γ がんま in 3 yields the horizontal lift of any vector field τ たう = τ たう μ みゅー ∂μ みゅー on X to a projectable vector field
Γ がんま
τ たう
=
τ たう
⌋
Γ がんま
=
τ たう
μ みゅー
(
∂
μ みゅー
+
Γ がんま
μ みゅー
i
∂
i
)
⊂
H
Y
{\displaystyle \Gamma \tau =\tau \rfloor \Gamma =\tau ^{\mu }\left(\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\right)\subset \mathrm {H} Y}
on Y .
Connection as a vertical-valued form [ edit ]
The horizontal splitting 2 of the exact sequence 1 defines the corresponding splitting of the dual exact sequence
0
→
Y
×
X
T
∗
X
→
T
∗
Y
→
V
∗
Y
→
0
,
{\displaystyle 0\to Y\times _{X}\mathrm {T} ^{*}X\to \mathrm {T} ^{*}Y\to \mathrm {V} ^{*}Y\to 0\,,}
where T*Y and T*X are the cotangent bundles of Y , respectively, and V*Y → Y is the dual bundle to VY → Y , called the vertical cotangent bundle. This splitting is given by the vertical-valued form
Γ がんま
=
(
d
y
i
−
Γ がんま
λ らむだ
i
d
x
λ らむだ
)
⊗
∂
i
,
{\displaystyle \Gamma =\left(dy^{i}-\Gamma _{\lambda }^{i}dx^{\lambda }\right)\otimes \partial _{i}\,,}
which also represents a connection on a fibered manifold.
Treating a connection as a vertical-valued form, one comes to the following important construction. Given a fibered manifold Y → X , let f : X ′ → X be a morphism and f ∗ Y → X ′ the pullback bundle of Y by f . Then any connection Γ がんま 3 on Y → X induces the pullback connection
f
∗
Γ がんま
=
(
d
y
i
−
(
Γ がんま
∘
f
~
)
λ らむだ
i
∂
f
λ らむだ
∂
x
′
μ みゅー
d
x
′
μ みゅー
)
⊗
∂
i
{\displaystyle f*\Gamma =\left(dy^{i}-\left(\Gamma \circ {\tilde {f}}\right)_{\lambda }^{i}{\frac {\partial f^{\lambda }}{\partial x'^{\mu }}}dx'^{\mu }\right)\otimes \partial _{i}}
on f ∗ Y → X ′ .
Connection as a jet bundle section [ edit ]
Let J1 Y be the jet manifold of sections of a fibered manifold Y → X , with coordinates (xμ みゅー , yi , yi μ みゅー ) . Due to the canonical imbedding
J
1
Y
→
Y
(
Y
×
X
T
∗
X
)
⊗
Y
T
Y
,
(
y
μ みゅー
i
)
→
d
x
μ みゅー
⊗
(
∂
μ みゅー
+
y
μ みゅー
i
∂
i
)
,
{\displaystyle \mathrm {J} ^{1}Y\to _{Y}\left(Y\times _{X}\mathrm {T} ^{*}X\right)\otimes _{Y}\mathrm {T} Y\,,\qquad \left(y_{\mu }^{i}\right)\to dx^{\mu }\otimes \left(\partial _{\mu }+y_{\mu }^{i}\partial _{i}\right)\,,}
any connection Γ がんま 3 on a fibered manifold Y → X is represented by a global section
Γ がんま
:
Y
→
J
1
Y
,
y
λ らむだ
i
∘
Γ がんま
=
Γ がんま
λ らむだ
i
,
{\displaystyle \Gamma :Y\to \mathrm {J} ^{1}Y\,,\qquad y_{\lambda }^{i}\circ \Gamma =\Gamma _{\lambda }^{i}\,,}
of the jet bundle J1 Y → Y , and vice versa . It is an affine bundle modelled on a vector bundle
(
Y
×
X
T
∗
X
)
⊗
Y
V
Y
→
Y
.
{\displaystyle \left(Y\times _{X}T^{*}X\right)\otimes _{Y}\mathrm {V} Y\to Y\,.}
(4 )
There are the following corollaries of this fact.
Connections on a fibered manifold Y → X make up an affine space modelled on the vector space of soldering forms
σ しぐま
=
σ しぐま
μ みゅー
i
d
x
μ みゅー
⊗
∂
i
{\displaystyle \sigma =\sigma _{\mu }^{i}dx^{\mu }\otimes \partial _{i}}
(5 )
on Y → X , i.e., sections of the vector bundle 4 . Connection coefficients possess the coordinate transformation law
Γ がんま
′
λ らむだ
i
=
∂
x
μ みゅー
∂
x
′
λ らむだ
(
∂
μ みゅー
y
′
i
+
Γ がんま
μ みゅー
j
∂
j
y
′
i
)
.
{\displaystyle {\Gamma '}_{\lambda }^{i}={\frac {\partial x^{\mu }}{\partial {x'}^{\lambda }}}\left(\partial _{\mu }{y'}^{i}+\Gamma _{\mu }^{j}\partial _{j}{y'}^{i}\right)\,.}
Every connection Γ がんま on a fibred manifold Y → X yields the first order differential operator
D
Γ がんま
:
J
1
Y
→
Y
T
∗
X
⊗
Y
V
Y
,
D
Γ がんま
=
(
y
λ らむだ
i
−
Γ がんま
λ らむだ
i
)
d
x
λ らむだ
⊗
∂
i
,
{\displaystyle D_{\Gamma }:\mathrm {J} ^{1}Y\to _{Y}\mathrm {T} ^{*}X\otimes _{Y}\mathrm {V} Y\,,\qquad D_{\Gamma }=\left(y_{\lambda }^{i}-\Gamma _{\lambda }^{i}\right)dx^{\lambda }\otimes \partial _{i}\,,}
on Y called the covariant differential relative to the connection Γ がんま . If s : X → Y is a section, its covariant differential
∇
Γ がんま
s
=
(
∂
λ らむだ
s
i
−
Γ がんま
λ らむだ
i
∘
s
)
d
x
λ らむだ
⊗
∂
i
,
{\displaystyle \nabla ^{\Gamma }s=\left(\partial _{\lambda }s^{i}-\Gamma _{\lambda }^{i}\circ s\right)dx^{\lambda }\otimes \partial _{i}\,,}
and the covariant derivative
∇
τ たう
Γ がんま
s
=
τ たう
⌋
∇
Γ がんま
s
{\displaystyle \nabla _{\tau }^{\Gamma }s=\tau \rfloor \nabla ^{\Gamma }s}
along a vector field τ たう on X are defined.
Curvature and torsion [ edit ]
Given the connection Γ がんま 3 on a fibered manifold Y → X , its curvature is defined as the Nijenhuis differential
R
=
1
2
d
Γ がんま
Γ がんま
=
1
2
[
Γ がんま
,
Γ がんま
]
F
N
=
1
2
R
λ らむだ
μ みゅー
i
d
x
λ らむだ
∧
d
x
μ みゅー
⊗
∂
i
,
R
λ らむだ
μ みゅー
i
=
∂
λ らむだ
Γ がんま
μ みゅー
i
−
∂
μ みゅー
Γ がんま
λ らむだ
i
+
Γ がんま
λ らむだ
j
∂
j
Γ がんま
μ みゅー
i
−
Γ がんま
μ みゅー
j
∂
j
Γ がんま
λ らむだ
i
.
{\displaystyle {\begin{aligned}R&={\tfrac {1}{2}}d_{\Gamma }\Gamma \\&={\tfrac {1}{2}}[\Gamma ,\Gamma ]_{\mathrm {FN} }\\&={\tfrac {1}{2}}R_{\lambda \mu }^{i}\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,,\\R_{\lambda \mu }^{i}&=\partial _{\lambda }\Gamma _{\mu }^{i}-\partial _{\mu }\Gamma _{\lambda }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\Gamma _{\mu }^{i}-\Gamma _{\mu }^{j}\partial _{j}\Gamma _{\lambda }^{i}\,.\end{aligned}}}
This is a vertical-valued horizontal two-form on Y .
Given the connection Γ がんま 3 and the soldering form σ しぐま 5 , a torsion of Γ がんま with respect to σ しぐま is defined as
T
=
d
Γ がんま
σ しぐま
=
(
∂
λ らむだ
σ しぐま
μ みゅー
i
+
Γ がんま
λ らむだ
j
∂
j
σ しぐま
μ みゅー
i
−
∂
j
Γ がんま
λ らむだ
i
σ しぐま
μ みゅー
j
)
d
x
λ らむだ
∧
d
x
μ みゅー
⊗
∂
i
.
{\displaystyle T=d_{\Gamma }\sigma =\left(\partial _{\lambda }\sigma _{\mu }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\sigma _{\mu }^{i}-\partial _{j}\Gamma _{\lambda }^{i}\sigma _{\mu }^{j}\right)\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,.}
Bundle of principal connections [ edit ]
Let π ぱい : P → M be a principal bundle with a structure Lie group G . A principal connection on P usually is described by a Lie algebra-valued connection one-form on P . At the same time, a principal connection on P is a global section of the jet bundle J1 P → P which is equivariant with respect to the canonical right action of G in P . Therefore, it is represented by a global section of the quotient bundle C = J1 P /G → M , called the bundle of principal connections . It is an affine bundle modelled on the vector bundle VP /G → M whose typical fiber is the Lie algebra g of structure group G , and where G acts on by the adjoint representation . There is the canonical imbedding of C to the quotient bundle TP /G which also is called the bundle of principal connections .
Given a basis {em } for a Lie algebra of G , the fiber bundle C is endowed with bundle coordinates (xμ みゅー , am μ みゅー ) , and its sections are represented by vector-valued one-forms
A
=
d
x
λ らむだ
⊗
(
∂
λ らむだ
+
a
λ らむだ
m
e
m
)
,
{\displaystyle A=dx^{\lambda }\otimes \left(\partial _{\lambda }+a_{\lambda }^{m}{\mathrm {e} }_{m}\right)\,,}
where
a
λ らむだ
m
d
x
λ らむだ
⊗
e
m
{\displaystyle a_{\lambda }^{m}\,dx^{\lambda }\otimes {\mathrm {e} }_{m}}
are the familiar local connection forms on M .
Let us note that the jet bundle J1 C of C is a configuration space of Yang–Mills gauge theory . It admits the canonical decomposition
a
λ らむだ
μ みゅー
r
=
1
2
(
F
λ らむだ
μ みゅー
r
+
S
λ らむだ
μ みゅー
r
)
=
1
2
(
a
λ らむだ
μ みゅー
r
+
a
μ みゅー
λ らむだ
r
−
c
p
q
r
a
λ らむだ
p
a
μ みゅー
q
)
+
1
2
(
a
λ らむだ
μ みゅー
r
−
a
μ みゅー
λ らむだ
r
+
c
p
q
r
a
λ らむだ
p
a
μ みゅー
q
)
,
{\displaystyle {\begin{aligned}a_{\lambda \mu }^{r}&={\tfrac {1}{2}}\left(F_{\lambda \mu }^{r}+S_{\lambda \mu }^{r}\right)\\&={\tfrac {1}{2}}\left(a_{\lambda \mu }^{r}+a_{\mu \lambda }^{r}-c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)+{\tfrac {1}{2}}\left(a_{\lambda \mu }^{r}-a_{\mu \lambda }^{r}+c_{pq}^{r}a_{\lambda }^{p}a_{\mu }^{q}\right)\,,\end{aligned}}}
where
F
=
1
2
F
λ らむだ
μ みゅー
m
d
x
λ らむだ
∧
d
x
μ みゅー
⊗
e
m
{\displaystyle F={\tfrac {1}{2}}F_{\lambda \mu }^{m}\,dx^{\lambda }\wedge dx^{\mu }\otimes {\mathrm {e} }_{m}}
is called the strength form of a principal connection.
See also [ edit ]
^ Krupka, Demeter; Janyška, Josef (1990). Lectures on differential invariants . Univerzita J. E. Purkyně v Brně. p. 174. ISBN 80-210-0165-8 .
References [ edit ]
Kolář, Ivan; Michor, Peter; Slovák, Jan (1993). Natural operators in differential geometry (PDF) . Springer-Verlag. Archived from the original (PDF) on 2017-03-30. Retrieved 2013-05-28 .
Krupka, Demeter; Janyška, Josef (1990). Lectures on differential invariants . Univerzita J. E. Purkyně v Brně. ISBN 80-210-0165-8 .
Saunders, D.J. (1989). The geometry of jet bundles . Cambridge University Press. ISBN 0-521-36948-7 .
Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory . World Scientific. ISBN 981-02-2013-8 .
Sardanashvily, G. (2013). Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory . Lambert Academic Publishing. arXiv :0908.1886 . Bibcode :2009arXiv0908.1886S . ISBN 978-3-659-37815-7 .