Connection (fibred manifold)

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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 $\Gamma :Y\to J^{1}Y$, where $J^{1}Y$ is the jet manifold of $Y$.

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 trough 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) $\theta _{X}=dx^{\mu }\otimes \partial _{\mu }$ on $X$, and vice versa. With this form, the horizontal splitting (2) reads

$\Gamma :\partial _{\lambda }\to \partial _{\lambda }\rfloor \Gamma =\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i}.$

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

$\Gamma \tau =\tau \rfloor \Gamma =\tau ^{\lambda }(\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i})\subset HY$

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

$\Gamma =(dy^{i}-\Gamma _{\lambda }^{i}dx^{\lambda })\otimes \partial _{i},$

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

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

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

Connection as a jet bundle section

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

$J^{1}Y\to _{Y}(Y\times _{X}T^{*}X)\otimes _{Y}TY,\qquad (y_{\mu }^{i})\to dx^{\mu }\otimes (\partial _{\mu }+y_{\mu }^{i}\partial _{i}),$

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

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

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

$(Y\times _{X}T^{*}X)\otimes _{Y}VY\to Y.\qquad \qquad (4)$

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

$\sigma =\sigma _{\mu }^{i}dx^{\mu }\otimes \partial _{i}\qquad \qquad (5)$

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

(ii) Connection coefficients possess the coordinate transformation law

${\Gamma '}_{\lambda }^{i}={\frac {\partial x^{\mu }}{\partial {x'}^{\lambda }}}(\partial _{\mu }{y'}^{i}+\Gamma _{\mu }^{j}\partial _{j}{y'}^{i}).$

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

$D_{\Gamma }:J^{1}Y\to _{Y}T^{*}X\otimes _{Y}VY,\qquad D_{\Gamma }=(y_{\lambda }^{i}-\Gamma _{\lambda }^{i})dx^{\lambda }\otimes \partial _{i},$

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

$R={\frac {1}{2}}d_{\Gamma }\Gamma ={\frac {1}{2}}[\Gamma ,\Gamma ]_{FN}={\frac {1}{2}}R_{\lambda \mu }^{i}\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i},$

$R_{\lambda \mu }^{i}=\partial _{\lambda }\Gamma _{\mu }^{i}-\partial _{\mu }\Gamma _{\lambda }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\Gamma _{\mu }^{i}-\Gamma _{\mu }^{j}\partial _{j}\Gamma _{\lambda }^{i}.$

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 $J^{1}P\to P$ 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 $VP/G\to M$ 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 $(x^{\mu },a_{\mu }^{m})$, and its sections are represented by vector-valued one-forms

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

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

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

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.

See also

• Connection (mathematics)
• Fibred manifold
• Ehresmann connection
• Connection (principal bundle)

References

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• 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