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 $\pi: Y\to X$ be a fibered manifold. A (generalized) connection on $Y$ is a section $\Gamma: Y\to J^1Y$ , where $J^1Y$ is the jet manifold of $Y$ .^[1]

Connection as a horizontal splitting

Let $\pi: Y\to X$ be a fibered manifold. There is the following canonical short exact sequence of vector bundles over $Y$ :

0\to VY\to TY\to Y\times_X TX\to 0, \qquad\qquad (1)

where $TY$ and $TX$ are the tangent bundles of $Y$ , respectively, $VY$ is the vertical tangent bundle of $Y$ , and $Y\times_X TX$ is the pullback bundle of $TX$ onto $Y$ .

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

\Gamma: Y\times_X TX \to TY \qquad\qquad (2)

over $Y$ which splits the exact sequence (1). A connection always exists.

Sometimes, this connection $\Gamma$ is called the Ehresmann connection because it yields the horizontal distribution

HY=\Gamma(Y\times_X TX ) \subset TY

of $TY$ and its horizontal decomposition $TY=VY\oplus HY$ .

At the same time, by an Ehresmann connection also is meant the following construction. Any connection $\Gamma$ on a fibered manifold $Y\to X$ yields a horizontal lift $\Gamma\circ\tau$ of a vector field $\tau$ on $X$ onto $Y$ , but need not defines the similar lift of a path in $X$ into $Y$ . Let $\mathbb R\supset[,]\ni t\to x(t)\in X$ and $\mathbb R\ni t\to y(t)\in Y$ be smooth paths in $X$ and $Y$ , respectively. Then $t\to y(t)$ is called the horizontal lift of $x(t)$ if $\pi(y(t))= x(t)$ , $\dot y(t)\in HY$ , $t\in {\mathbb R}$ . A connection $\Gamma$ is said to be the Ehresmann connection if, for each path $x([0,1])$ in $X$ , there exists its horizontal lift through any point $y\in\pi^{-1}(x([0,1]))$ . 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 $(x^\mu, y^i)$ , and let $\Gamma$ be a connection on $Y\to X$ . It yields uniquely the horizontal tangent-valued one-form

\Gamma = dx^\lambda\otimes (\partial_\lambda + \Gamma_\lambda^i(x^\nu, y^j)\partial_i) \qquad\qquad (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_\lambda\to \partial_\lambda\rfloor\Gamma=\partial_\lambda +\Gamma^i_\lambda\partial_i.

In particular, the connection $\Gamma$ (3) yields the horizontal lift of any vector field $\tau=\tau^\mu\partial_\mu$ on $X$ to a projectable vector field

\Gamma \tau=\tau\rfloor\Gamma=\tau^\lambda(\partial_\lambda +\Gamma^i_\lambda\partial_i)\subset HY

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 T^*X \to T^*Y\to 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 $VY\to Y$ , called the vertical cotangent bundle. This splitting is given by the vertical-valued form

\Gamma= (dy^i -\Gamma^i_\lambda dx^\lambda)\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 $\Gamma$ (3) on $Y\to X$ induces the pullback connection

f^*\Gamma=(dy^i-(\Gamma\circ \widetilde f)^i_\lambda\frac{\partial f^\lambda}{\partial x'^\mu}dx'^\mu)\otimes\partial_i

on $f^*Y\to X'$ .

Connection as a jet bundle section

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

J^1Y\to_Y (Y\times_X T^*X )\otimes_Y TY, \qquad (y^i_\mu)\to dx^\mu\otimes (\partial_\mu + y^i_\mu\partial_i),

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

\Gamma :Y\to J^1Y, \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

(Y\times_X T^*X )\otimes_Y VY\to Y. \qquad\qquad (4)

There are the following corollaries of this fact.

(i) Connections on a fibered manifold $Y\to X$ make up an affine space modelled on the vector space of soldering forms

\sigma=\sigma^i_\mu dx^\mu\otimes\partial_i \qquad\qquad (5)

on $Y\to X$ , i.e., sections of the vector bundle (4).

(ii) Connection coefficients possess the coordinate transformation law

{\Gamma'}^i_\lambda = \frac{\partial x^\mu}{\partial {x'}^\lambda}(\partial_\mu {y'}^i +\Gamma^j_\mu\partial_j{y'}^i).

(iii) Every connection $\Gamma$ on a fibred manifold $Y\to X$ yields the first order differential operator

D_\Gamma:J^1Y\to_Y T^*X\otimes_Y VY, \qquad D_\Gamma =(y^i_\lambda -\Gamma^i_\lambda)dx^\lambda\otimes\partial_i,

on $Y$ called the covariant differential relative to the connection $\Gamma$ . If $s:X\to Y$ is a section, its covariant differential

\nabla^\Gamma s = (\partial_\lambda s^i - \Gamma_\lambda^i\circ s) dx^\lambda\otimes \partial_i,

and the covariant derivative $\nabla_\tau^\Gamma s=\tau\rfloor\nabla^\Gamma s$ along a vector field $\tau$ on $X$ are defined.

Curvature and torsion

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

R=\frac{1}{2} d_\Gamma\Gamma=\frac{1}{2} [\Gamma,\Gamma]_{FN} = \frac12 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.

This is a vertical-valued horizontal two-form on $Y$ .

Given the connection $\Gamma$ (3) and the soldering form $\sigma$ (5), a torsion of $\Gamma$ with respect to $\sigma$ is defined as

T = d_\Gamma \sigma = (\partial_\lambda\sigma_\mu^i + \Gamma_\lambda^j\partial_j\sigma_\mu^i - \partial_j\Gamma_\lambda^i\sigma_\mu^j) \, dx^\lambda\wedge dx^\mu\otimes \partial_i.

Bundle of principal connections

Let $\pi\colon P\to M$ 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 $J^1P\to P$ 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 $C=J^1P/G\to M$ , called the bundle of principal connections. It is an affine bundle modelled on the vector bundle $VP/G\to M$ whose typical fiber is the Lie algebra ${\mathfrak {g}}$ 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 $\{{\mathrm e}_m\}$ for a Lie algebra of $G$ , the fiber bundle $C$ is endowed with bundle coordinates $(x^\mu,a^m_\mu)$ , and its sections are represented by vector-valued one-forms

A=dx^\lambda\otimes (\partial_\lambda + a^m_\lambda {\mathrm e}_m),

where $a^m_\lambda \, dx^\lambda\otimes {\mathrm e}_m$ are the familiar local connection forms on $M$ .

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

a_{\lambda\mu}^r = \frac12(F_{\lambda\mu}^r + S_{\lambda\mu}^r)= \frac{1}{2}(a_{\lambda\mu}^r + a_{\mu\lambda}^r - c_{pq}^r a_\lambda^p a_\mu^q) + \frac{1}{2} (a_{\lambda\mu}^r - a_{\mu\lambda}^r + c_{pq}^r a_\lambda^p a_\mu^q),

where

F=\frac{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, ISBN 80-210-0165-8 (page 174)

References

Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag
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., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN 981-02-2013-8.

External links

Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ISBN 978-3-659-37815-7; arXiv: 0908.1886

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