Connection (affine bundle) - Revision history

Idea at 22:14, 29 September 2026

2026-09-29T22:14:19Z

← Older revision		Revision as of 22:14, 29 September 2026
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	~~{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}~~
	}}
	~~{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}~~

	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''}}.		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''}}.

Waza: remove deletion notice

2026-09-29T21:45:27Z

remove deletion notice

← Older revision		Revision as of 21:45, 29 September 2026
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	~~<!-- Please do not remove or change this AfD message until the discussion has been closed. -->~~
	~~{{Article for deletion/dated\|page=Connection (affine bundle)\|timestamp=20260924190147\|year=2026\|month=September\|day=24\|substed=yes\|help=off\|outcome=merging\|target=Affine bundle}}~~
	~~<!-- Once discussion is closed, please place on talk page: {{Old AfD multi\|page=Connection (affine bundle)\|date=24 September 2026\|result='''keep'''}} -->~~
	~~<!-- End of AfD message, feel free to edit beyond this point -->~~
	{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}		{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}
	}}		}}

Waza: 25 revisions imported

2026-09-29T21:44:06Z

25 revisions imported

← Older revision	Revision as of 21:44, 29 September 2026
(No difference)

wikipedia>Adkbeckett: Nominated for merging; see :Wikipedia:Articles for deletion/Connection (affine bundle).

2026-09-24T19:01:47Z

Nominated for merging; see Wikipedia:Articles for deletion/Connection (affine bundle).

← Older revision		Revision as of 19:01, 24 September 2026
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			<!-- Please do not remove or change this AfD message until the discussion has been closed. -->
			{{Article for deletion/dated\|page=Connection (affine bundle)\|timestamp=20260924190147\|year=2026\|month=September\|day=24\|substed=yes\|help=off\|outcome=merging\|target=Affine bundle}}
			<!-- Once discussion is closed, please place on talk page: {{Old AfD multi\|page=Connection (affine bundle)\|date=24 September 2026\|result='''keep'''}} -->
			<!-- End of AfD message, feel free to edit beyond this point -->
	{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}		{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}
	}}		}}

wikipedia>Adkbeckett: Cancelled PROD.

2026-09-24T18:48:17Z

Cancelled PROD.

← Older revision		Revision as of 18:48, 24 September 2026
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	~~{{Proposed deletion/dated~~
	\|concern = This article is a stub written almost entirely by a single editor in 2013, and it appears to be copied near-verbatim from a single source from around the same time, likely written by the editor themself (comparing the editor's username to the name of the source's author). Thus there are elements of [[WP:SELFCITE]] and [[WP:NOR]] if not [[WP:PLAG]]. As an aside, if the content is even notable enough to be retained, it would fit better as a subsection of [[Affine bundle]].
	~~\|timestamp = 20260923191731~~
	~~\|nom = Adkbeckett~~
	~~\|help = off~~
	}}
	{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}		{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}
	}}		}}

wikipedia>Adkbeckett: typo

2026-09-24T16:42:54Z

typo

← Older revision		Revision as of 16:42, 24 September 2026
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	{{Proposed deletion/dated		{{Proposed deletion/dated
	\|concern = This article is a stub written almost entirely by a single editor in 2013, and it appears to be copied near-verbatim from a single source from around the same time, likely written by the editor themself (comparing the editor's username to the name of the source's author). Thus there are elements of [[WP:SELFCITE]] and [[WP:NOR]] if not [[WP:PLAG]]. As an aside, if the content is even notable enough to be retained, it would fit better as a ~~subection~~ of [[Affine bundle]].		\|concern = This article is a stub written almost entirely by a single editor in 2013, and it appears to be copied near-verbatim from a single source from around the same time, likely written by the editor themself (comparing the editor's username to the name of the source's author). Thus there are elements of [[WP:SELFCITE]] and [[WP:NOR]] if not [[WP:PLAG]]. As an aside, if the content is even notable enough to be retained, it would fit better as a subsection of [[Affine bundle]].
	\|timestamp = 20260923191731		\|timestamp = 20260923191731
	\|nom = Adkbeckett		\|nom = Adkbeckett

wikipedia>Jean Raimbault: slight change to lede

2026-09-24T16:40:31Z

slight change to lede

← Older revision		Revision as of 16:40, 24 September 2026
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	{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}		{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}

	In [[differential geometry]], ~~the classical notion of~~ a [[~~connection (mathematics)~~]]~~, most commonly encountered in~~ the ~~context~~ of [[~~Connection (vector bundle)\|vector bundles]], can be generalised to [[affine~~ bundle]]s. 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''}}.		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''}}.

	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.		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.
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	: <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>		: <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 [[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 ~~[[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\|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		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

wikipedia>Adkbeckett: Proposing article for deletion per WP:PROD.

2026-09-23T19:17:31Z

Proposing article for deletion per WP:PROD.

← Older revision		Revision as of 19:17, 23 September 2026
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			{{Proposed deletion/dated
			\|concern = This article is a stub written almost entirely by a single editor in 2013, and it appears to be copied near-verbatim from a single source from around the same time, likely written by the editor themself (comparing the editor's username to the name of the source's author). Thus there are elements of [[WP:SELFCITE]] and [[WP:NOR]] if not [[WP:PLAG]]. As an aside, if the content is even notable enough to be retained, it would fit better as a subection of [[Affine bundle]].
			\|timestamp = 20260923191731
			\|nom = Adkbeckett
			\|help = off
			}}
	{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}		{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}
	}}		}}

wikipedia>Adkbeckett: Deleted irrelevant reference. The reference is a standard one for differential geometry, but does not discuss connections on an affine bundle as defined here (only connections on the affine frame bundle of the tangent bundle).

2026-09-23T19:02:55Z

Deleted irrelevant reference. The reference is a standard one for differential geometry, but does not discuss connections on an affine bundle as defined here (only connections on the affine frame bundle of the tangent bundle).

← Older revision		Revision as of 19:02, 23 September 2026
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	==References==		==References==
	* {{cite book\|first1=S. \|last1=Kobayashi \|first2=K. \|last2=Nomizu \|author-link=Shoshichi Kobayashi\|author2-link=Katsumi Nomizu \|title=[[Foundations of Differential Geometry]] \|volume=1–2 \|publisher=Wiley-Interscience \|date=1996 \|isbn=0-471-15733-3}}
	* {{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 }}		* {{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 }}

wikipedia>Adkbeckett: Added an introductory sentence and a distinguish tag, cleaned up the lead paragraph.

2026-09-23T18:59:03Z

Added an introductory sentence and a distinguish tag, cleaned up the lead paragraph.

← Older revision		Revision as of 18:59, 23 September 2026
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	{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}		{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}
	}}		}}
			{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}

	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 ~~the~~ '''affine connection''' if it 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''}} is an affine bundle morphism over {{math\|''X''}}. ~~In particular,~~ this ~~is an~~ [[affine connection]] on the [[tangent bundle]] {{math\|T''X''}} of a [[smooth manifold]] {{math\|''X''}}~~. (That is~~, the ~~connection on an affine bundle is an~~ example ~~of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use~~ of the ~~adjective "affine"~~.)		In [[differential geometry]], the classical notion of a [[connection (mathematics)]], most commonly encountered in the context of [[Connection (vector bundle)\|vector bundles]], can be generalised to [[affine bundle]]s. 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''}}.

			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.

	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]]		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]]

← Older revision		Revision as of 16:40, 24 September 2026
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	{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}		{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}

	In [[differential geometry]], ~~the classical notion of~~ a [[~~connection (mathematics)~~]]~~, most commonly encountered in~~ the ~~context~~ of [[~~Connection (vector bundle)\|vector bundles]], can be generalised to [[affine~~ bundle]]s. 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''}}.		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''}}.

	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.		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.
Line 17:		Line 17:
	: <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>		: <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 [[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 ~~[[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\|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		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

← Older revision		Revision as of 18:59, 23 September 2026
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	{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}		{{Multiple issues\|{{expert-subject\|1=Mathematics\|date=October 2013\|reason=There is no lead and it needs to be better organized.}}{{no footnotes\|date=October 2013}}
	}}		}}
			{{distinguish\|text=[[Affine connection]], a connection on the tangent bundle of a smooth manifold.}}

	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 ~~the~~ '''affine connection''' if it 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''}} is an affine bundle morphism over {{math\|''X''}}. ~~In particular,~~ this ~~is an~~ [[affine connection]] on the [[tangent bundle]] {{math\|T''X''}} of a [[smooth manifold]] {{math\|''X''}}~~. (That is~~, the ~~connection on an affine bundle is an~~ example ~~of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use~~ of the ~~adjective "affine"~~.)		In [[differential geometry]], the classical notion of a [[connection (mathematics)]], most commonly encountered in the context of [[Connection (vector bundle)\|vector bundles]], can be generalised to [[affine bundle]]s. 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''}}.

			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.

	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]]		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]]