Connection (fibred manifold)

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

Connection as a horizontal splitting

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

${\displaystyle 0\to VY\to TY\to Y\times _{X}TX\to 0,\qquad \qquad (1)}$

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

${\displaystyle \Gamma :Y\times _{X}TX\to TY\qquad \qquad (2)}$

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

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

${\displaystyle HY=\Gamma (Y\times _{X}TX)\subset TY}$

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

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

${\displaystyle \Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{i}(x^{\nu },y^{j})\partial _{i})\qquad \qquad (3)}$

on ${\displaystyle Y}$ which projects onto the canonical tangent-valued form (tautological one-form) ${\displaystyle \theta _{X}=dx^{\mu }\otimes \partial _{\mu }}$ on ${\displaystyle X}$, and vice versa. With this form, the horizontal splitting (2) reads

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

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

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

Connection as a vertical-valued form

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

${\displaystyle 0\to Y\times _{X}T^{*}X\to T^{*}Y\to V^{*}Y\to 0,}$

where ${\displaystyle T^{*}Y}$ and ${\displaystyle T^{*}X}$ are the cotangent bundles of ${\displaystyle Y}$, respectively, and ${\displaystyle V^{*}Y\to Y}$ is the dual bundle to ${\displaystyle VY\to Y}$, called the vertical cotangent bundle. This splitting is given by the vertical-valued form

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

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 ${\displaystyle Y\to X}$, let ${\displaystyle f:X'\to X}$ be a morphism and ${\displaystyle f^{*}Y\to X'}$ the pullback bundle of ${\displaystyle Y}$ by ${\displaystyle f}$. Then any connection ${\displaystyle \Gamma }$ (3) on ${\displaystyle Y\to X}$ induces the pullback connection

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

Connection as a jet bundle section

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

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

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

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

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

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

There are the following corollaries of this fact.

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

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

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

(ii) Connection coefficients possess the coordinate transformation law

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

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

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

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

${\displaystyle \nabla ^{\Gamma }s=(\partial _{\lambda }s^{i}-\Gamma _{\lambda }^{i}\circ s)dx^{\lambda }\otimes \partial _{i},}$

Curvature and torsion

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

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

This is a vertical-valued horizontal two-form on ${\displaystyle Y}$.

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

${\displaystyle T=d_{\Gamma }\sigma =(\partial _{\lambda }\sigma _{\mu }^{i}+\Gamma _{\lambda }^{j}\partial _{j}\sigma _{\mu }^{i}-\partial _{j}\Gamma _{\lambda }^{i}\sigma _{\mu }^{j})\,dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}.}$

Bundle of principal connections

Let ${\displaystyle \pi \colon P\to M}$ be a principal bundle with a structure Lie group ${\displaystyle G}$. A principal connection on ${\displaystyle P}$ usually is described by a Lie algebra-valued connection one-form on ${\displaystyle P}$. At the same time, a principal connection on ${\displaystyle P}$ is a global section of the jet bundle ${\displaystyle J^{1}P\to P}$ which is equivariant with respect to the canonical right action of ${\displaystyle G}$ in ${\displaystyle P}$. Therefore, it is represented by a global section of the quotient bundle ${\displaystyle C=J^{1}P/G\to M}$, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle ${\displaystyle VP/G\to M}$ whose typical fiber is the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of structure group ${\displaystyle G}$, and where ${\displaystyle G}$ acts on by the adjoint representation. There is the canonical imbedding of ${\displaystyle C}$ to the quotient bundle ${\displaystyle TP/G}$ which also is called the bundle of principal connections.

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

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

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

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

where

${\displaystyle F={\frac {1}{2}}F_{\lambda \mu }^{m}\,dx^{\lambda }\wedge dx^{\mu }\otimes {\mathrm {e} }_{m}}$

is called the strength form of a principal connection.

Notes

1. {{#invoke:citation/CS1|citation |CitationClass=citation }} (page 174)

References

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• Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN 981-02-2013-8.