Formal definition Connection (fibred manifold)

1 formal definition

1.1 connection horizontal splitting
1.2 connection tangent-valued form
1.3 connection vertical-valued form
1.4 connection jet bundle section

formal definition

let π : y → x fibered manifold. generalized connection on y section Γ : y → jy, jy jet manifold of y.

connection horizontal splitting

with above manifold π there following canonical short exact sequence of vector bundles on y:

where ty , tx tangent bundles of y, respectively, vy vertical tangent bundle of y, , y ×x tx pullback bundle of tx onto y.

a connection on fibered manifold y → x defined linear bundle morphism

over y splits exact sequence 1. connection exists.

sometimes, connection Γ called ehresmann connection because yields horizontal distribution

h

y
=
Γ

(
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 , horizontal decomposition ty = vy ⊕ hy.

at same time, ehresmann connection meant following construction. connection Γ on fibered manifold y → x yields horizontal lift Γ ∘ τ of vector field τ on x onto y, need not defines similar lift of path in x 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 2 smooth paths in x , y, respectively. t → y(t) called horizontal lift of x(t) if

π
(
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 said ehresmann connection if, each path x([0,1]) in x, there exists horizontal lift through point y ∈ π(x([0,1])). fibered manifold fiber bundle if , if admits such ehresmann connection.

connection tangent-valued form

given fibered manifold y → x, let endowed atlas of fibered coordinates (x, y), , let Γ connection on y → x. yields uniquely horizontal tangent-valued one-form

on y projects onto canonical tangent-valued form (tautological one-form or solder form)

θ

x


=
d

x

μ


⊗

∂

μ




{\displaystyle \theta _{x}=dx^{\mu }\otimes \partial _{\mu }}

on x, , vice versa. form, horizontal splitting 2 reads

Γ
:

∂

λ


→

∂

λ


⌋
Γ
=

∂

λ


+

Γ

λ


i



∂

i



.


{\displaystyle \gamma :\partial _{\lambda }\to \partial _{\lambda }\rfloor \gamma =\partial _{\lambda }+\gamma _{\lambda }^{i}\partial _{i}\,.}

in particular, connection Γ in 3 yields horizontal lift of vector field τ = τ ∂μ on x projectable vector field

Γ
τ
=
τ
⌋
Γ
=

τ

λ



(

∂

λ


+

Γ

λ


i



∂

i


)

⊂

h

y


{\displaystyle \gamma \tau =\tau \rfloor \gamma =\tau ^{\lambda }\left(\partial _{\lambda }+\gamma _{\lambda }^{i}\partial _{i}\right)\subset \mathrm {h} y}

on y.

connection vertical-valued form

the horizontal splitting 2 of exact sequence 1 defines corresponding splitting of 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 , t*x cotangent bundles of y, respectively, , v*y → y dual bundle vy → y, called vertical cotangent bundle. splitting given vertical-valued form

Γ
=

(
d

y

i


−

Γ

λ


i


d

x

λ


)

⊗

∂

i



,


{\displaystyle \gamma =\left(dy^{i}-\gamma _{\lambda }^{i}dx^{\lambda }\right)\otimes \partial _{i}\,,}

which represents connection on fibered manifold.

treating connection vertical-valued form, 1 comes following important construction. given fibered manifold y → x, let f : x′ → x morphism , f ∗ y → x′ pullback bundle of y f. connection Γ 3 on y → x induces pullback connection

f
∗
Γ
=

(
d

y

i


−


(
Γ
∘



f
~



)


λ


i





∂

f

λ




∂

x

′

μ






d

x

′

μ



)

⊗

∂

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 jet bundle section

let jy jet manifold of sections of fibered manifold y → x, coordinates (x, y, yi

μ). due canonical imbedding

j


1


y

→

y



(
y

×

x




t


∗


x
)


⊗

y



t

y

,


(

y

μ


i


)

→
d

x

μ


⊗

(

∂

μ


+

y

μ


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 3 on fibered manifold y → x represented global section

Γ
:
y
→


j


1


y

,


y

λ


i


∘
Γ
=

Γ

λ


i



,


{\displaystyle \gamma :y\to \mathrm {j} ^{1}y\,,\qquad y_{\lambda }^{i}\circ \gamma =\gamma _{\lambda }^{i}\,,}

of jet bundle jy → y, , vice versa. affine bundle modelled on vector bundle

there following corollaries of fact.