Connection (fibred manifold)

Short description: Operation on fibered manifolds

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 $Y$ :

[math]\displaystyle{ 0\to \mathrm{V}Y\to \mathrm{T}Y\to Y\times_X \mathrm{T}X\to 0\,, }[/math]

(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

[math]\displaystyle{ \Gamma: Y\times_X \mathrm{T}X \to \mathrm{T}Y }[/math]

(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

[math]\displaystyle{ \mathrm{H}Y=\Gamma\left(Y\times_X \mathrm{T}X \right) \subset \mathrm{T}Y }[/math]

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

[math]\displaystyle{ \begin{align}\mathbb R\supset[,]\ni t&\to x(t)\in X \\ \mathbb R\ni t&\to y(t)\in Y\end{align} }[/math]

be two smooth paths in $X$ and $Y$ , respectively. Then $t \to y (t)$ is called the horizontal lift of $x (t)$ if

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

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

[math]\displaystyle{ \Gamma = dx^\mu\otimes \left(\partial_\mu + \Gamma_\mu^i\left(x^\nu, y^j\right)\partial_i\right) }[/math]

(3)

on $Y$ which projects onto the canonical tangent-valued form (tautological one-form or solder form)

[math]\displaystyle{ \theta_X=dx^\mu\otimes\partial_\mu }[/math]

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

[math]\displaystyle{ \Gamma:\partial_\mu\to \partial_\mu\rfloor\Gamma=\partial_\mu +\Gamma^i_\mu\partial_i\,. }[/math]

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

[math]\displaystyle{ \Gamma \tau=\tau\rfloor\Gamma=\tau^\mu\left(\partial_\mu +\Gamma^i_\mu\partial_i\right)\subset \mathrm{H}Y }[/math]

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

[math]\displaystyle{ 0\to Y\times_X \mathrm{T}^*X \to \mathrm{T}^*Y\to \mathrm{V}^*Y\to 0\,, }[/math]

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

[math]\displaystyle{ \Gamma= \left(dy^i -\Gamma^i_\lambda dx^\lambda\right)\otimes\partial_i\,, }[/math]

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

[math]\displaystyle{ f*\Gamma=\left(dy^i-\left(\Gamma\circ \tilde f\right)^i_\lambda\frac{\partial f^\lambda}{\partial x'^\mu}dx'^\mu\right)\otimes\partial_i }[/math]

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

[math]\displaystyle{ \mathrm{J}^1Y\to_Y \left(Y\times_X \mathrm{T}^*X \right)\otimes_Y \mathrm{T}Y\,, \qquad \left(y^i_\mu\right)\to dx^\mu\otimes \left(\partial_\mu + y^i_\mu\partial_i\right)\,, }[/math]

any connection $Γ$ 3 on a fibered manifold $Y \to X$ is represented by a global section

[math]\displaystyle{ \Gamma :Y\to \mathrm{J}^1Y\,, \qquad y_\lambda^i\circ\Gamma=\Gamma_\lambda^i\,, }[/math]

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

[math]\displaystyle{ \left(Y\times_X T^*X \right)\otimes_Y \mathrm{V}Y\to Y\,. }[/math]

(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

[math]\displaystyle{ \sigma=\sigma^i_\mu dx^\mu\otimes\partial_i }[/math]

(5)

on $Y \to X$ , i.e., sections of the vector bundle 4.
Connection coefficients possess the coordinate transformation law
[math]\displaystyle{ {\Gamma'}^i_\lambda = \frac{\partial x^\mu}{\partial {x'}^\lambda}\left(\partial_\mu {y'}^i+\Gamma^j_\mu\partial_j{y'}^i\right)\,. }[/math]
Every connection $Γ$ on a fibred manifold $Y \to X$ yields the first order differential operator
[math]\displaystyle{ D_\Gamma:\mathrm{J}^1Y\to_Y \mathrm{T}^*X\otimes_Y \mathrm{V}Y\,, \qquad D_\Gamma = \left(y^i_\lambda -\Gamma^i_\lambda\right)dx^\lambda\otimes\partial_i\,, }[/math]
on $Y$ called the covariant differential relative to the connection $Γ$ . If $s : X \to Y$ is a section, its covariant differential
[math]\displaystyle{ \nabla^\Gamma s = \left(\partial_\lambda s^i - \Gamma_\lambda^i\circ s\right) dx^\lambda\otimes \partial_i\,, }[/math]
and the covariant derivative
[math]\displaystyle{ \nabla_\tau^\Gamma s=\tau\rfloor\nabla^\Gamma s }[/math]
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

[math]\displaystyle{ \begin{align} R&=\tfrac12 d_\Gamma\Gamma\\&=\tfrac12 [\Gamma,\Gamma]_\mathrm{FN} \\&= \tfrac12 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{align} }[/math]

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

[math]\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\,. }[/math]

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

[math]\displaystyle{ A=dx^\lambda\otimes \left(\partial_\lambda + a^m_\lambda {\mathrm e}_m\right)\,, }[/math]

where

[math]\displaystyle{ a^m_\lambda \, dx^\lambda\otimes {\mathrm e}_m }[/math]

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

[math]\displaystyle{ \begin{align} a_{\lambda\mu}^r &= \tfrac12\left(F_{\lambda\mu}^r + S_{\lambda\mu}^r\right) \\ &= \tfrac12\left(a_{\lambda\mu}^r + a_{\mu\lambda}^r - c_{pq}^r a_\lambda^p a_\mu^q\right) + \tfrac12\left(a_{\lambda\mu}^r - a_{\mu\lambda}^r + c_{pq}^r a_\lambda^p a_\mu^q\right)\,, \end{align} }[/math]

where

[math]\displaystyle{ F=\tfrac{1}{2} F_{\lambda\mu}^m \, dx^\lambda\wedge dx^\mu\otimes {\mathrm e}_m }[/math]

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. Springer-Verlag. http://www.emis.de/monographs/KSM/kmsbookh.pdf. 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. https://archive.org/details/geometryofjetbun0000saun.
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. ISBN 978-3-659-37815-7. Bibcode: 2009arXiv0908.1886S.

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

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Formal definition

Connection as a horizontal splitting

Connection as a tangent-valued form

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Connection (fibred manifold)

Formal definition

Connection as a horizontal splitting

Connection as a tangent-valued form

Connection as a vertical-valued form

Connection as a jet bundle section

Curvature and torsion

Bundle of principal connections

See also

Notes

References

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