Levi-Civita connection - Revision history

Vipul: /* Christoffel symbols */

2012-01-06T17:59:04Z

Christoffel symbols

← Older revision		Revision as of 17:59, 6 January 2012
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	{{further\|[[Christoffel symbols of a connection]]}}		{{further\|[[Christoffel symbols of a connection]]}}

	The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times ~~T_p~~(M) \to ~~T_p~~(M)</math>, which roughly ''differentiates'' one tangent vector along another.		The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times \Gamma(TM) \to \Gamma(TM)</math>, which roughly ''differentiates'' one tangent vector along another.

	Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_{ij}^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>.		Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_{ij}^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>.

	The Christoffel symbols thus give an ''explicit description'' of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.		The Christoffel symbols thus give an ''explicit description'' of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.

Vipul at 21:02, 24 July 2009

2009-07-24T21:02:06Z

← Older revision		Revision as of 21:02, 24 July 2009
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	<math>g(\nabla_XY,Z) = \frac{Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])}{2}</math>.		<math>g(\nabla_XY,Z) = \frac{Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])}{2}</math>.

			===Additional use===

			The term '''Levi-Civita connection''' is sometimes also used for the connection induced on any tensor product involving the tangent and cotangent bundle, using the rule for [[tensor product of connections]] and the [[dual connection]].

	==Facts==		==Facts==

Vipul: /* Facts */

2009-07-24T20:54:48Z

Facts

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	The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of <math>g</math>.		The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of <math>g</math>.

	Note that the nondegeneracy of <math>g</math> is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of <math>nabla_XY</math>		Note that the nondegeneracy of <math>g</math> is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of <math>\nabla_XY</math>

	~~To show that the Levi-Civita connection exists, it suffices to check that the map sending <math>Z</math> to what we propose for <math>g(\nabla_X Y, Z)</math> is actually a linear map.~~

	===Christoffel symbols===		===Christoffel symbols===

Vipul: /* Christoffel symbols */

2009-07-24T20:54:16Z

Christoffel symbols

← Older revision		Revision as of 20:54, 24 July 2009
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	===Christoffel symbols===		===Christoffel symbols===

	{{further\|[[Christoffel ~~symbol~~]]}}		{{further\|[[Christoffel symbols of a connection]]}}

	The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another.		The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another.

Vipul at 19:53, 24 July 2009

2009-07-24T19:53:07Z

← Older revision		Revision as of 19:53, 24 July 2009
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	* The connection is [[defining ingredient::metric connection\|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>		* The connection is [[defining ingredient::metric connection\|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
	* The connection is [[defining ingredient::torsion-free linear connection\|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>		* The connection is [[defining ingredient::torsion-free linear connection\|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>.

			===Definition by formula===

			The formula for the Levi-Civita connection requires us to use the fact that <math>g</math> is nondegenerate. So, instead of directly specifying <math>\nabla_XY</math> for vector fields <math>X</math> and <math>Y</math>, the formula specifies <math>g(\nabla_XY,Z)</math> for <math>X,Y,Z</math> vector fields, as follows:

			<math>g(\nabla_XY,Z) = \frac{Xg(Y,Z) + Yg(Z,X) - Zg(X,Y) + g(Y,[Z,X]) + g(Z,[X,Y]) - g(X,[Y,Z])}{2}</math>.

	==Facts==		==Facts==

Vipul at 16:13, 24 July 2009

2009-07-24T16:13:22Z

← Older revision		Revision as of 16:13, 24 July 2009
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	===Given data===		===Given data===

	A [[Riemannian 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 [[Riemannian manifold]] <math>(M,g)</math> (here, <math>M</math> is a [[differential manifold]] and <math>g</math> is the additional structure of a [[defining ingredient::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 bilinear form <math>g</math> in each tangent space.		More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a [[defining ingredient::pseudo-Riemannian metric]]: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form <math>g</math> in each tangent space.

	===Definition part===		===Definition part===

	A '''Levi-Civita connection''' on <math>(M,g)</math> is a [[linear connection]] <math>\nabla</math> on <math>M</math> satisfying the following two conditions:		A '''Levi-Civita connection''' on <math>(M,g)</math> is a [[defining ingredient::linear connection]] <math>\nabla</math> on <math>M</math> satisfying the following two conditions:

	* The connection is [[metric connection\|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>		* The connection is [[defining ingredient::metric connection\|metric]], viz <math>X g (Y,Z) = g (\nabla_X Y, Z) + g(Y, \nabla_X Z)</math>
	* The connection is [[torsion-free linear connection\|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>		* The connection is [[defining ingredient::torsion-free linear connection\|torsion-free]], viz <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>

	==Facts==		==Facts==
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	===The Levi-Civita connection is unique===		===The Levi-Civita connection is unique===

	~~The proof of the uniqueness of the~~ Levi-Civita connection is ~~as follows.~~		{{further\|[[Levi-Civita connection exists and is unique]]}}

	* Take three vector fields <math>X, Y, Z</math>. Now, consider the ~~three equations obtained by cycling <math>X, Y, Z</math> in the first~~ condition~~. Solving this system of linear equations~~, ~~we can express <math>g(\nabla_XY,Z)</math> in terms of <math>X,Y,Z</math>.~~		The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of <math>g</math>.

	~~Explicitly:~~

	~~<math>g(\nabla_XY,Z) + g(Y,\nabla_XZ) = Xg(Y,Z)</math>~~

	~~<math>g(\nabla_YX,Z) + g(X,\nabla_YZ) = Yg(Z,X)</math>~~

	~~<math>g(\nabla_ZX,Y) + g(X,\nabla_ZY) = Zg(X,Y)</math>~~

	~~Now let's choose to focus only on the clockwise cyclic expressions, that is, the three expressions <math>p = g(\nabla_XY,Z), q = g(\nabla_YZ,X), q = g(\nabla_ZX,Y)</math>.~~

	~~Writing everything in terms of these three (we here make use of~~ the torsion ~~tensor vanishing):~~

	~~<math>p + q = Xg(Y,Z)~~ - ~~g(Y,[X,Z])</math>~~

	~~and similarly for the other three variables.~~

	~~We thus get expressions for <math>g(\nabla_XY~~,~~Z)</math> in terms of <math>g</math>~~ and the ~~Lie bracket.~~

	* Now use the fact that <math>g</math> is nondegenerate to conclude that knowledge of ~~the function~~ <math>g~~(\nabla_XY,Z)</math> for ''all'' <math>Z</math> helps us fix <math>\nabla_XY</math> uniquely.~~
	* This means that we have a unique definition for <math>\nabla</math>.

	Note that the nondegeneracy of <math>g</math> is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of <math>nabla_XY</math>		Note that the nondegeneracy of <math>g</math> is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of <math>nabla_XY</math>

Vipul: 8 revisions

2008-05-18T19:48:05Z

8 revisions

← Older revision	Revision as of 19:48, 18 May 2008
(No difference)

Vipul: /* Christoffel symbols */

2007-09-02T02:39:13Z

Christoffel symbols

← Older revision		Revision as of 02:39, 2 September 2007
Line 50:		Line 50:

	===Christoffel symbols===		===Christoffel symbols===

			{{further\|[[Christoffel symbol]]}}

	The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another.		The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another.

Vipul: /* Given data */

2007-08-30T12:31:35Z

Given data

← Older revision		Revision as of 12:31, 30 August 2007
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	A [[Riemannian 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 [[Riemannian 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.		More generally, we can also look at a [[pseudo-Riemannian manifold]], or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form <math>g</math> in each tangent space.

	===Definition part===		===Definition part===

Vipul at 12:28, 30 August 2007

2007-08-30T12:28:27Z

← Older revision		Revision as of 12:28, 30 August 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 [[Riemannian 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.		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.