Connection (fibred manifold)

In differential geometry, a fibered manifold is surjective submersion of smooth manifolds $Y \to X$ . 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

Let $π : Y \to X$ be a fibered manifold. A generalized connection on $Y$ is a section $Γ : Y \to J 1 Y$ , where $J 1 Y$ is the jet manifold of $Y$ .^[1]

Connection as a horizontal splitting

With the above manifold $π$ there is the following canonical short exact sequence of vector bundles over $Y$ :

0\to \mathrm {V} Y\to \mathrm {T} Y\to Y\times _{X}\mathrm {T} X\to 0\,,

(1)

where $T Y$ and $T X$ are the tangent bundles of $Y$ , respectively, $V Y$ is the vertical tangent bundle of $Y$ , and $Y \times X T X$ is the pullback bundle of $T X$ onto $Y$ .

A connection on a fibered manifold $Y \to X$ is defined as a linear bundle morphism

\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 $Γ$ is called the Ehresmann connection because it yields the horizontal distribution

\mathrm {H} Y=\Gamma \left(Y\times _{X}\mathrm {T} X\right)\subset \mathrm {T} Y

of $T Y$ and its horizontal decomposition $T Y = V Y \oplus H Y$ .

At the same time, by an Ehresmann connection also is meant the following construction. Any connection $Γ$ on a fibered manifold $Y \to X$ yields a horizontal lift $Γ \circ τ$ of a vector field $τ$ on $X$ onto $Y$ , but need not defines the similar lift of a path in $X$ into $Y$ . Let

{\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 $t \to y (t)$ is called the horizontal lift of $x (t)$ if

\pi (y(t))=x(t)\,,\qquad {\dot {y}}(t)\in \mathrm {H} Y\,,\qquad t\in \mathbb {R} \,.

A connection $Γ$ is said to be the Ehresmann connection if, for each path $x ([0,1])$ in $X$ , there exists its horizontal lift through any point $y \in π -1 (x ([0,1]))$ . A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.

Connection as a tangent-valued form

Given a fibered manifold $Y \to X$ , let it be endowed with an atlas of fibered coordinates $(x μ, y i)$ , and let $Γ$ be a connection on $Y \to X$ . It yields uniquely the horizontal tangent-valued one-form

\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)

\theta _{X}=dx^{\mu }\otimes \partial _{\mu }

on $X$ , and vice versa. With this form, the horizontal splitting 2 reads

\Gamma :\partial _{\mu }\to \partial _{\mu }\rfloor \Gamma =\partial _{\mu }+\Gamma _{\mu }^{i}\partial _{i}\,.

In particular, the connection $Γ$ in 3 yields the horizontal lift of any vector field $τ = τ μ \partial μ$ on $X$ to a projectable vector field

\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

The horizontal splitting 2 of the exact sequence 1 defines the corresponding splitting of the dual exact sequence

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 $V* Y \to Y$ is the dual bundle to $V Y \to Y$ , called the vertical cotangent bundle. This splitting is given by the vertical-valued form

\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 $Y \to X$ , let $f : X' \to X$ be a morphism and $f * Y \to X'$ the pullback bundle of $Y$ by $f$ . Then any connection $Γ$ 3 on $Y \to X$ induces the pullback connection

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 $f * Y \to X'$ .

Connection as a jet bundle section

Let $J 1 Y$ be the jet manifold of sections of a fibered manifold $Y \to X$ , with coordinates $(x μ, y i, y i μ)$ . Due to the canonical imbedding

\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 a fibered manifold $Y \to X$ is represented by a global section

\Gamma :Y\to \mathrm {J} ^{1}Y\,,\qquad y_{\lambda }^{i}\circ \Gamma =\Gamma _{\lambda }^{i}\,,

of the jet bundle $J 1 Y \to Y$ , and vice versa. It is an affine bundle modelled on a vector bundle

\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 \to X

make up an affine space modelled on the vector space of soldering forms

\sigma =\sigma _{\mu }^{i}dx^{\mu }\otimes \partial _{i}

(5)

on

Y \to X

, i.e., sections of the vector bundle 4.

Connection coefficients possess the coordinate transformation law
${\Gamma '}_{\lambda }^{i}={\frac {\partial x^{\mu }}{\partial {x'}^{\lambda }}}\left(\partial _{\mu }{y'}^{i}+\Gamma _{\mu }^{j}\partial _{j}{y'}^{i}\right)\,.$
Every connection $Γ$ on a fibred manifold $Y \to X$ yields the first order differential operator
$D_{\Gamma }:\mathrm {J} ^{1}Y\to _{Y}\mathrm {T} ^{*}X\otimes _{Y}\mathrm {V} Y\,,\qquad D_{\Gamma }=\left(y_{\lambda }^{i}-\Gamma _{\lambda }^{i}\right)dx^{\lambda }\otimes \partial _{i}\,,$
on $Y$ called the covariant differential relative to the connection $Γ$ . If $s : X \to Y$ is a section, its covariant differential
$\nabla ^{\Gamma }s=\left(\partial _{\lambda }s^{i}-\Gamma _{\lambda }^{i}\circ s\right)dx^{\lambda }\otimes \partial _{i}\,,$
and the covariant derivative
$\nabla _{\tau }^{\Gamma }s=\tau \rfloor \nabla ^{\Gamma }s$
along a vector field $τ$ on $X$ are defined.

Curvature and torsion

Given the connection $Γ$ 3 on a fibered manifold $Y \to X$ , its curvature is defined as the Nijenhuis differential

{\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 $Γ$ 3 and the soldering form $σ$ 5, a torsion of $Γ$ with respect to $σ$ is defined as

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

Let $π : P \to M$ 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 $J 1 P \to P$ 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 $C = J 1 P / G \to M$ , called the bundle of principal connections. It is an affine bundle modelled on the vector bundle $V P / G \to M$ 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 $T P / G$ which also is called the bundle of principal connections.

Given a basis ${e m$ } for a Lie algebra of $G$ , the fiber bundle $C$ is endowed with bundle coordinates $(x μ, a m μ)$ , and its sections are represented by vector-valued one-forms

A=dx^{\lambda }\otimes \left(\partial _{\lambda }+a_{\lambda }^{m}{\mathrm {e} }_{m}\right)\,,

where

a_{\lambda }^{m}\,dx^{\lambda }\otimes {\mathrm {e} }_{m}

are the familiar local connection forms on $M$ .

Let us note that the jet bundle $J 1 C$ of $C$ is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition

{\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={\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.

Notes

^ Krupka, Demeter; Janyška, Josef (1990). Lectures on differential invariants. Univerzita J. E. Purkyně v Brně. p. 174. ISBN 80-210-0165-8.

References

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.

[1] Krupka, Demeter; Janyška, Josef (1990). Lectures on differential invariants. Univerzita J. E. Purkyně v Brně. p. 174. ISBN 80-210-0165-8.

[1]