Pointwise - Revision history

84.150.115.60: /* Componentwise operations */ K can not be generalized to a set since on a set, there is no addition and thats what this section is about

2014-12-09T11:02:53Z

Componentwise operations: K can not be generalized to a set since on a set, there is no addition and thats what this section is about

← Older revision		Revision as of 13:02, 9 December 2014
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	In [[mathematics]], a '''metric connection''' is a [[connection (vector bundle)\|connection]] in a [[vector bundle]] ''E'' equipped with a [[metric (vector bundle)\|metric]]<!--Red link until someone wants to write an appropriate article. ~~[[metric tensor]] isn't right~~.~~--> for which the [[inner product]] of any two vectors will remain the same when those vectors are [[parallel transport]]ed along any curve~~. ~~Other common equivalent formulations of a metric connection include:~~		Hello friend. Let me introduce myself. I am Luther Aubrey. Interviewing is what she does but soon she'll be on her personal. Kansas is our beginning place and my parents reside nearby. What she enjoys doing is taking part in croquet and she is trying to make it a occupation.<br><br>Feel free to visit my web page: [http://www.madgamers.eu/index/users/view/id/7632 extended car warranty]
	* A connection for which the [[connection (vector bundle)\|covariant derivative]]s of the metric on '~~'E'' vanish.~~
	* A [[connection (principal bundle)\|principal connection]] on ~~the bundle of [[orthonormal frame]]s of ''E''~~.

	~~A special case of a metric connection~~ is ~~the [[Levi-Civita connection]]~~. ~~Here the bundle ''E''~~ is ~~the [[tangent bundle]] of a manifold. In addition to being a metric connection, the Levi-Civita connection~~ is ~~required~~ to ~~be [[torsion tensor\|torsion free]].~~

	~~==Riemannian connections==~~
	~~An important special case of a metric connection is~~ a ~~'''Riemannian connection'''~~. ~~This is a connection~~ <~~math~~>~~\nabla~~<~~/math~~> on the [[tangent bundle]] of a [[pseudo-Riemannian manifold]] (''M'', ''g'') such that <math>\nabla_X g = 0</math> for all vector fields ''X'' on ''M''. Equivalently, <math>\nabla</math> is Riemannian if the [[parallel transport]] it defines preserves the metric ''g''.

	~~A given connection <math>\nabla</math> is Riemannian if and only if~~
	:~~<math>Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ) </math>~~
	~~for all vector fields ''X'', ''Y'' and ''Z'' on ''M'', where <math>Xg(Y,Z)</math> denotes the derivative of the function <math>g(Y,Z)</math> along this vector field <math>X</math>.~~

	~~The [[Levi-Civita connection]] is the [[torsion tensor\|torsion-free]] Riemannian connection on a manifold. It is unique by the [[fundamental theorem of Riemannian geometry]].~~

	~~==External links==~~
	*[http://~~projecteuclid~~.~~org/Dienst~~/UI/~~1.0~~/~~Summarize~~/~~euclid.cmp~~/~~1103858479 a pdf about this]~~

	~~[[Category:Connection (mathematics)]]~~
	~~[[Category:Riemannian geometry]~~]


	~~{{differential-geometry-stub}}~~

en>EmausBot: Bot: Migrating 3 langlinks, now provided by Wikidata on d:Q9248237

2013-04-02T10:54:25Z

Bot: Migrating 3 langlinks, now provided by Wikidata on d:Q9248237

← Older revision		Revision as of 12:54, 2 April 2013
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	~~Hello friend~~. ~~Allow me introduce myself~~. ~~I am Luther Aubrey~~. ~~Delaware~~ is the ~~location I adore most but I need~~ to ~~move for my family~~. ~~She~~ is ~~presently~~ a ~~cashier but soon she~~'~~ll be~~ on ~~her own~~. ~~To play badminton~~ is ~~something he really enjoys performing~~.<br><br>~~Also visit my web-site; [http~~://~~www~~.~~neonex~~.~~cl/Default~~.~~aspx?tabid~~=~~61&userId~~=~~4739~~ http://~~www~~.~~neonex~~.cl/]		In [[mathematics]], a '''metric connection''' is a [[connection (vector bundle)\|connection]] in a [[vector bundle]] ''E'' equipped with a [[metric (vector bundle)\|metric]]<!--Red link until someone wants to write an appropriate article. [[metric tensor]] isn't right.--> for which the [[inner product]] of any two vectors will remain the same when those vectors are [[parallel transport]]ed along any curve. Other common equivalent formulations of a metric connection include:
			* A connection for which the [[connection (vector bundle)\|covariant derivative]]s of the metric on ''E'' vanish.
			* A [[connection (principal bundle)\|principal connection]] on the bundle of [[orthonormal frame]]s of ''E''.

			A special case of a metric connection is the [[Levi-Civita connection]]. Here the bundle ''E'' is the [[tangent bundle]] of a manifold. In addition to being a metric connection, the Levi-Civita connection is required to be [[torsion tensor\|torsion free]].

			==Riemannian connections==
			An important special case of a metric connection is a '''Riemannian connection'''. This is a connection <math>\nabla</math> on the [[tangent bundle]] of a [[pseudo-Riemannian manifold]] (''M'', ''g'') such that <math>\nabla_X g = 0</math> for all vector fields ''X'' on ''M''. Equivalently, <math>\nabla</math> is Riemannian if the [[parallel transport]] it defines preserves the metric ''g''.

			A given connection <math>\nabla</math> is Riemannian if and only if
			:<math>Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ) </math>
			for all vector fields ''X'', ''Y'' and ''Z'' on ''M'', where <math>Xg(Y,Z)</math> denotes the derivative of the function <math>g(Y,Z)</math> along this vector field <math>X</math>.

			The [[Levi-Civita connection]] is the [[torsion tensor\|torsion-free]] Riemannian connection on a manifold. It is unique by the [[fundamental theorem of Riemannian geometry]].

			==External links==
			*[http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103858479 a pdf about this]

			[[Category:Connection (mathematics)]]
			[[Category:Riemannian geometry]]


			{{differential-geometry-stub}}

en>Beroal: /* Pointwise operations */

2012-07-17T21:37:46Z

Pointwise operations

New page

Hello friend. Allow me introduce myself. I am Luther Aubrey. Delaware is the location I adore most but I need to move for my family. She is presently a cashier but soon she'll be on her own. To play badminton is something he really enjoys performing.<br><br>Also visit my web-site; [http://www.neonex.cl/Default.aspx?tabid=61&userId=4739 http://www.neonex.cl/]