Metric connection - Revision history

Vipul: 4 revisions

2008-05-18T19:48:44Z

4 revisions

← Older revision	Revision as of 19:48, 18 May 2008
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Vipul: /* Definition */

2007-09-02T06:59:01Z

Definition

← Older revision		Revision as of 06:59, 2 September 2007
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	===Given data===		===Given data===

	A [[~~Riemmanian~~ manifold]] <math>(M~~,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g~~</math> ~~is the additional structure of a [[Riemannian metric]] on it)~~.		A [[differential manifold]] <math>M</math>.

	A ~~vector~~ bundle <math>E</math> over <math>M</math>, with a smoothly varying metric structure on ~~eac hfibre~~ of <math>E</math> over <math>M</math>.		A [[metric bundle]] <math>E</math> over <math>M</math> (viz, a [[vector bundle]] with a smoothly varying metric structure <math>g</math> on each fibre of <math>E</math> over <math>M</math>).

	===Definition part===		===Definition part===
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	In other words, it is a connection such that the dual connection on the dual bundle to <math>E</math> is the same as the connection obtained by the natural isomorphism between <math>E</math> and its dual (induced by the metric).		In other words, it is a connection such that the dual connection on the dual bundle to <math>E</math> is the same as the connection obtained by the natural isomorphism between <math>E</math> and its dual (induced by the metric).

	We are in particular interested in [[metric linear connection]]s, which are metric connections over the tangent bundle. Of these, a very special one is the [[Levi-Civita connection]], which is the only [[torsion-free linear connection\|torsion-free]] metric linear connection.		We are in particular interested in [[metric linear connection]]s, which are metric connections over the tangent bundle for a [[Riemannian manifold]] (viz, the tangent bundle is endowed with a [[Riemannian metric]]). Of these, a very special one is the [[Levi-Civita connection]], which is the only [[torsion-free linear connection\|torsion-free]] metric linear connection.

Vipul at 11:39, 23 June 2007

2007-06-23T11:39:30Z

← Older revision		Revision as of 11:39, 23 June 2007
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	A [[Riemmanian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[Riemannian metric]] on it).		A [[Riemmanian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[Riemannian metric]] on it).

	A vector bundle <math>E</math> over <math>M</math>.		A vector bundle <math>E</math> over <math>M</math>, with a smoothly varying metric structure on eac hfibre of <math>E</math> over <math>M</math>.
	~~More generally, we can also look at a [[pseudo-Riemannian manifold]]~~, ~~or a manifold~~ with a smoothly varying ~~nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space~~.
	===Definition part===		===Definition part===

	A '''metric connection''' on <math>(M,g)</math> is a [[~~linear~~ connection]] <math>\nabla</math> on <math>M</math> satisfying the following condition:		A '''metric connection''' on <math>(M,g)</math> is a [[connection]] <math>\nabla</math> on the vector bundle <math>E</math> over <math>M</math> satisfying the following condition:

	<math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>		<math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>

	In other words, it is a connection such that the dual connection on the ~~cotangent~~ bundle is the same as the connection obtained by the natural isomorphism between ~~the tangent~~ and ~~cotangent bundle~~ (induced by the metric).		Here <math>X</math> is a vector field (viz a section of <math>TM</math>) and <math>Y</math> and <math>Z</math> are sections of <math>E</math>.

			In other words, it is a connection such that the dual connection on the dual bundle to <math>E</math> is the same as the connection obtained by the natural isomorphism between <math>E</math> and its dual (induced by the metric).

			We are in particular interested in [[metric linear connection]]s, which are metric connections over the tangent bundle. Of these, a very special one is the [[Levi-Civita connection]], which is the only [[torsion-free linear connection\|torsion-free]] metric linear connection.

Vipul at 11:34, 23 June 2007

2007-06-23T11:34:18Z

← Older revision		Revision as of 11:34, 23 June 2007
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	{{elementof\|the space of all [[~~linear~~ connection]]s, which in turn sits inside the space of all <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>}}		{{elementof\|the space of all [[connection]]s, which in turn sits inside the space of all <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>}}

	{{connection property}}		{{connection property}}
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	A [[Riemmanian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[Riemannian metric]] on it).		A [[Riemmanian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[Riemannian metric]] on it).

			A vector bundle <math>E</math> over <math>M</math>.
	More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space.		More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space.
	===Definition part===		===Definition part===

Vipul at 11:33, 23 June 2007

2007-06-23T11:33:06Z

New page

{{elementof|the space of all [[linear connection]]s, which in turn sits inside the space of all <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>}}

{{connection property}}

==Definition==

===Given data===

A [[Riemmanian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[Riemannian metric]] on it).

More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space.
===Definition part===

A '''metric connection''' on <math>(M,g)</math> is a [[linear connection]] <math>\nabla</math> on <math>M</math> satisfying the following condition:

<math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>

In other words, it is a connection such that the dual connection on the cotangent bundle is the same as the connection obtained by the natural isomorphism between the tangent and cotangent bundle (induced by the metric).