Connection (affine bundle): Difference between revisions

Latest revision as of 22:14, 29 September 2026

In differential geometry, a connection on an affine bundle is a specialisation to affine bundles of the more general notion of a connection on a principal bundle. Let Template:Math be an affine bundle modelled over a vector bundle Template:Math. A connection Template:Math on Template:Math is called an affine connection if, as a section Template:Math of the jet bundle Template:Math of Template:Math, it is an affine bundle morphism over Template:Math.

The term "affine connection" as used in this article should not be confused with its more common usage, namely a connection on the tangent bundle Template:Math of a smooth manifold Template:Math, though as discussed below, the latter can be considered as a special example of the former.

With respect to affine bundle coordinates Template:Math on Template:Math, an affine connection Template:Math on Template:Math is given by the tangent-valued connection form

<math>\begin{align}\Gamma &=dx^\lambda\otimes \left(\partial_\lambda + \Gamma_\lambda^i\partial_i\right)\,, \\ \Gamma_\lambda^i&={{\Gamma_\lambda}^i}_j\left(x^\nu\right) y^j + \sigma_\lambda^i\left(x^\nu\right)\,. \end{align}</math>

An affine bundle is a fiber bundle with a general affine structure group Template:Math of affine transformations of its typical fiber Template:Math of dimension Template:Math. Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection Template:Math, the corresponding linear derivative Template:Math of an affine morphism Template:Math defines a unique linear connection on a vector bundle Template:Math. With respect to linear bundle coordinates Template:Math on Template:Math, this connection reads

<math> \overline \Gamma=dx^\lambda\otimes\left(\partial_\lambda +{{\Gamma_\lambda}^i}_j\left(x^\nu\right) \overline y^j\overline\partial_i\right)\,.</math>

Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.

If Template:Math is a vector bundle, both an affine connection Template:Math and an associated linear connection Template:Math are connections on the same vector bundle Template:Math, and their difference is a basic soldering form on

<math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,.</math>

Thus, every affine connection on a vector bundle Template:Math is a sum of a linear connection and a basic soldering form on Template:Math.

Due to the canonical vertical splitting Template:Math, this soldering form is brought into a vector-valued form

<math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i </math>

where Template:Math is a fiber basis for Template:Math.

Given an affine connection Template:Math on a vector bundle Template:Math, let Template:Math and Template:Math be the curvatures of a connection Template:Math and the associated linear connection Template:Math, respectively. It is readily observed that Template:Math, where

<math>\begin{align}

T &=\tfrac12 T_{\lambda\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i\,, \\ T_{\lambda \mu}^i &= \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h {{\Gamma_\mu}^i}_h - \sigma_\mu^h {{\Gamma_\lambda}^i}_h\,, \end{align}</math>

is the torsion of Template:Math with respect to the basic soldering form Template:Math.

In particular, consider the tangent bundle Template:Math of a manifold Template:Math coordinated by Template:Math. There is the canonical soldering form

<math>\theta=dx^\mu\otimes \dot\partial_\mu </math>

on Template:Math which coincides with the tautological one-form

<math>\theta_X=dx^\mu\otimes \partial_\mu</math>

on Template:Math due to the canonical vertical splitting Template:Math. Given an arbitrary linear connection Template:Math on Template:Math, the corresponding affine connection

<math>\begin{align}

A&=\Gamma +\theta\,, \\ A_\lambda^\mu&={{\Gamma_\lambda}^\mu}_\nu \dot x^\nu +\delta^\mu_\lambda\,, \end{align}</math>

on Template:Math is the Cartan connection. The torsion of the Cartan connection Template:Math with respect to the soldering form Template:Math coincides with the torsion of a linear connection Template:Math, and its curvature is a sum Template:Math of the curvature and the torsion of Template:Math.

References

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.

Template:Differential-geometry-stub

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-{{Multiple issues|{{expert-subject|date=October 2013|reason=There is no lead and it needs to be organized more.}}{{cleanup|date=October 2013|reason=There is no lead and it needs to be organized more.}}{{no footnotes|date=October 2013}}
+In [[differential geometry]], a '''connection on an affine bundle''' is a specialisation to [[affine bundle]]s of the more general notion of a [[connection on a principal bundle]]. Let {{math|''Y'' → ''X''}} be an affine bundle modelled over a vector bundle {{math|{{overline|''Y''}} → ''X''}}. A [[connection (fibred manifold)|connection]] {{math|Γ}} on {{math|''Y'' → ''X''}} is called an '''affine connection''' if, as a section {{math|Γ : ''Y'' → J<sup>1</sup>''Y''}} of the [[jet bundle]] {{math|J<sup>1</sup>''Y'' → ''Y''}} of {{math|''Y''}}, it is an affine bundle morphism over {{math|''X''}}.
-{{Lead missing|date=May 2013}}
-}}
-Let <math>Y\to X,</math> be an [[affine bundle]] modelled over a vector bundle
+The term "affine connection" as used in this article should not be confused with [[affine connection|its more common usage]], namely a connection on the [[tangent bundle]] {{math|T''X''}} of a [[smooth manifold]] {{math|''X''}}, though as discussed below, the latter can be considered as a special example of the former.
-<math>\overline Y\to X </math>. A [[connection (fibred manifold)|connection]] <math>\Gamma</math> on <math>Y\to X</math> is called the '''affine connection''' if it as a section <math>\Gamma:Y\to J^1Y</math> of the [[jet bundle]] <math>J^1Y\to Y</math> of <math>Y</math> is an affine bundle morphism over <math>X</math>. In particular, this is the case of an [[affine connection]] on the [[tangent bundle]] <math>TX</math> of a [[smooth manifold]] <math>X</math>.
-With respect to affine bundle coordinates <math>(x^\lambda,y^i)</math> on <math>Y</math>, an affine connection <math>\Gamma</math> on <math>Y\to X</math> is given by the [[connection (fibred manifold)|tangent-valued connection form]]
+With respect to affine bundle coordinates {{math|(''x<sup>λ</sup>'', ''y<sup>i</sup>'')}} on {{math|''Y''}}, an affine connection {{math|Γ}} on {{math|''Y'' → ''X''}} is given by the [[connection (fibred manifold)|tangent-valued connection form]]
-: <math>\Gamma =dx^\lambda\otimes (\partial_\lambda + \Gamma_\lambda^i\partial_i), \qquad
+: <math>\begin{align}\Gamma &=dx^\lambda\otimes \left(\partial_\lambda + \Gamma_\lambda^i\partial_i\right)\,, \\ \Gamma_\lambda^i&={{\Gamma_\lambda}^i}_j\left(x^\nu\right) y^j + \sigma_\lambda^i\left(x^\nu\right)\,. \end{align}</math>
-\Gamma_\lambda^i=\Gamma_\lambda{}^i{}_j(x^\nu) y^j + \sigma_\lambda^i(x^\nu). </math>
-An affine bundle is a fiber bundle with a [[affine group|general affine]] [[fiber bundle|structure group]] <math> GA(m,\mathbb R) </math> of affine transformations of its typical fiber <math>V</math> of dimension <math>m</math>. Therefore, an affine connection is associated to a [[connection (principal bundle)|principal connection]]. It always exists.
+An affine bundle is a fiber bundle with a [[affine group|general affine]] [[fiber bundle|structure group]] {{math|GA(''m'', ℝ)}} of affine transformations of its typical fiber {{math|''V''}} of dimension {{math|''m''}}. Therefore, an affine connection is associated to a principal connection. It always exists.
-For any affine connection <math>\Gamma:Y\to J^1Y</math>, the corresponding [[affine bundle|linear derivative]] <math>\overline\Gamma:\overline Y\to J^1\overline Y</math> of an affine morphism <math>\Gamma</math> defines a
+For any affine connection {{math|Γ : ''Y'' → J<sup>1</sup>''Y''}}, the corresponding [[affine bundle|linear derivative]]  {{math|{{overline|Γ}} : {{overline|''Y''}} → J<sup>1</sup>{{overline|''Y''}}}} of an affine morphism {{math|Γ}} defines a unique [[connection (vector bundle)|linear connection]] on a vector bundle {{math|{{overline|''Y''}} → ''X''}}. With respect to linear bundle coordinates {{math|(''x<sup>λ</sup>'', {{overline|''y''}}<sup>''i''</sup>)}} on {{math|{{overline|''Y''}}}}, this connection reads
-unique [[connection (vector bundle)|linear connection]] on a vector bundle <math>\overline Y\to X</math>. With respect to linear bundle
-coordinates  <math>(x^\lambda,\overline y^i)</math> on <math>\overline Y</math>, this connection reads
-: <math> \overline \Gamma=dx^\lambda\otimes(\partial_\lambda +\Gamma_\lambda{}^i{}_j(x^\nu) \overline y^j\overline\partial_i).</math>
+: <math> \overline \Gamma=dx^\lambda\otimes\left(\partial_\lambda +{{\Gamma_\lambda}^i}_j\left(x^\nu\right) \overline y^j\overline\partial_i\right)\,.</math>
 Since every vector bundle is an affine bundle, any linear connection on
 a vector bundle also is an affine connection.
-If <math>Y\to X</math> is a vector bundle, both an affine connection <math>\Gamma</math>
+If {{math|''Y'' → ''X''}} is a vector bundle, both an affine connection {{math|Γ}} and an associated linear connection {{math|{{overline|Γ}}}} are
-and an associated linear connection <math>\overline\Gamma</math> are
+connections on the same vector bundle {{math|''Y'' → ''X''}}, and their difference is a basic soldering form on
-connections on the same vector bundle <math>Y\to X</math>, and their
+: <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,.</math>
-difference is a basic soldering form on <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i </math>. Thus, every affine
+Thus, every affine connection on a vector bundle {{math|''Y'' → ''X''}} is a sum of a linear connection and a basic soldering form on {{math|''Y'' → ''X''}}.
-connection on a vector bundle <math>Y\to X</math> is a sum of a linear
-connection and a basic soldering form on <math>Y\to X</math>.
-It should be noted that, due to the canonical vertical splitting <math>VY=Y\times Y</math>, this soldering form is brought into a [[vector-valued differential form|vector-valued form]] <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i </math> where <math>e_i</math> is a fiber basis for <math>Y</math>.
+Due to the canonical vertical splitting {{math|V''Y'' {{=}} ''Y'' × ''Y''}}, this soldering form is brought into a [[vector-valued differential form|vector-valued form]]
+: <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i </math>
+where {{math|''e<sub>i</sub>''}} is a fiber basis for {{math|''Y''}}.
-Given an affine connection <math>\Gamma</math> on a vector bundle <math>Y\to X</math>, let <math>R</math> and <math>\overline R</math> be the [[connection (fibred manifold)|curvatures]] of a connection <math>\Gamma</math> and the associated linear connection <math>\overline \Gamma</math>, respectively.  It is readily observed that <math>R = \overline R + T</math>, where
+Given an affine connection {{math|Γ}} on a vector bundle {{math|''Y'' → ''X''}}, let {{math|''R''}} and {{math|{{overline|''R''}}}} be the [[connection (fibred manifold)|curvatures]] of a connection {{math|Γ}} and the associated linear connection {{math|{{overline|Γ}}}}, respectively.  It is readily observed that {{math|''R'' {{=}} {{overline|''R''}} + ''T''}}, where
-: <math>T =\frac12 T_{\lambda
+: <math>\begin{align}
-\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i, \qquad  T_{\lambda \mu}^i = \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h
+T &=\tfrac12 T_{\lambda\mu}^i dx^\lambda\wedge dx^\mu\otimes \partial_i\,, \\
-\Gamma_\mu{}^i{}_h - \sigma_\mu^h \Gamma_\lambda{}^i{}_h, </math>
+T_{\lambda \mu}^i &= \partial_\lambda\sigma_\mu^i - \partial_\mu\sigma_\lambda^i + \sigma_\lambda^h {{\Gamma_\mu}^i}_h - \sigma_\mu^h {{\Gamma_\lambda}^i}_h\,,
+\end{align}</math>
-is the [[connection (fibred manifold)|torsion]] of <math>\Gamma</math> with respect to the basic soldering form <math>\sigma</math>.
+is the [[connection (fibred manifold)|torsion]] of {{math|Γ}} with respect to the basic soldering form {{math|''σ''}}.
-In particular, let us consider the tangent bundle <math>TX</math> of a manifold <math>X</math> coordinated by <math>(x^\mu,\dot x^\mu)</math>. There is the canonical soldering form <math>\theta=dx^\mu\otimes \dot\partial_\mu </math> on <math>TX</math> which coincides with the [[tautological one-form]] <math>\theta_X=dx^\mu\otimes \partial_\mu</math> on <math>X</math> due to the canonical vertical splitting <math>VTX=TX\times TX</math>. Given an arbitrary linear connection <math>\Gamma</math> on <math>TX</math>, the corresponding affine connection
+In particular, consider the tangent bundle {{math|T''X''}} of a manifold {{math|''X''}} coordinated by {{math|(''x<sup>μ</sup>'', ''ẋ<sup>μ</sup>'')}}. There is the canonical soldering form
+:<math>\theta=dx^\mu\otimes \dot\partial_\mu </math>
+on {{math|T''X''}} which coincides with the [[tautological one-form]]
+:<math>\theta_X=dx^\mu\otimes \partial_\mu</math>
+on {{math|''X''}} due to the canonical vertical splitting {{math|VT''X'' {{=}} T''X'' × T''X''}}. Given an arbitrary linear connection {{math|Γ}} on {{math|T''X''}}, the corresponding affine connection
-: <math>A=\Gamma +\theta, \qquad
+: <math>\begin{align}
-A_\lambda^\mu=\Gamma_\lambda{}^\mu{}_\nu \dot x^\nu +\delta^\mu_\lambda, </math>
+A&=\Gamma +\theta\,, \\
+A_\lambda^\mu&={{\Gamma_\lambda}^\mu}_\nu \dot x^\nu +\delta^\mu_\lambda\,,
+\end{align}</math>
-on <math>TX</math> is the [[Cartan connection]]. The torsion of the Cartan
+on {{math|T''X''}} is the [[Cartan connection]]. The torsion of the Cartan connection {{math|''A''}} with respect to the soldering form {{math|''θ''}} coincides with the [[torsion tensor|torsion]] of a linear connection {{math|Γ}}, and its curvature is a sum {{math|''R'' + ''T''}} of the curvature and the torsion of {{math|Γ}}.
-connection <math>A</math> with respect to the soldering form <math>\theta</math> coincides with the [[torsion tensor|torsion]] of a linear connection <math>\Gamma</math>, and its curvature is a sum <math>R+T</math> of the curvature and the torsion of <math>\Gamma</math>.
-==References==
-* S. Kobayashi, K. Nomizu, ''Foundations of Differential Geometry'', Vols. 1 & 2, Wiley-Interscience, 1996, ISBN 0-471-15733-3.
-* [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theor'', Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; [http://xxx.lanl.gov/abs/0908.1886 arXiv: 0908.1886].
 ==See also==
@@ Line 57: / Line 53: @@
 *[[Connection (mathematics)]]
 *[[Affine gauge theory]]
+==References==
+* {{cite book|authorlink=Gennadi Sardanashvily|last=Sardanashvily |first=G. |title=Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory |publisher=Lambert Academic Publishing |date=2013 |isbn=978-3-659-37815-7 |arxiv=0908.1886|bibcode=2009arXiv0908.1886S }}
 [[Category:Differential geometry]]
 [[Category:Connection (mathematics)]]
+{{differential-geometry-stub}}

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Latest revision as of 22:14, 29 September 2026

See also

References