First Bianchi identity - Revision history

Vipul at 01:14, 24 July 2009

2009-07-24T01:14:25Z

← Older revision		Revision as of 01:14, 24 July 2009
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	Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[fact about::Levi-Civita connection]] arising from a [[Riemannian metric]] or [[pseudo-Riemannian metric]].		Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[fact about::Levi-Civita connection]] arising from a [[Riemannian metric]] or [[pseudo-Riemannian metric]].

			==Related facts==

			* [[Second Bianchi identity]] (also called the ''differential Bianchi identity'').
			* [[Curvature is tensorial]]
			* [[Torsion is tensorial]]
	==Proof==		==Proof==

Vipul at 01:13, 24 July 2009

2009-07-24T01:13:17Z

← Older revision		Revision as of 01:13, 24 July 2009
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	==Statement==		==Statement==

	Let <math>\nabla</math> be a [[torsion-free linear connection]]. The [[Riemann curvature tensor]] <math>R</math> of <math>\nabla</math> satisfies the following '''first Bianchi identity''' or '''algebraic Bianchi identity''':		Let <math>\nabla</math> be a [[fact about::torsion-free linear connection]]. The [[fact about::Riemann curvature tensor]] <math>R</math> of <math>\nabla</math> satisfies the following '''first Bianchi identity''' or '''algebraic Bianchi identity''':

	<math>R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0</math>		<math>R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0</math>
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	for any three vector fields <math>X,Y,Z</math>.		for any three vector fields <math>X,Y,Z</math>.

	Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[Levi-Civita connection]] arising from a [[Riemannian metric]] or [[pseudo-Riemannian metric]].		Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[fact about::Levi-Civita connection]] arising from a [[Riemannian metric]] or [[pseudo-Riemannian metric]].

	==Proof==		==Proof==

Vipul: 2 revisions

2008-05-18T19:40:07Z

2 revisions

← Older revision	Revision as of 19:40, 18 May 2008
(No difference)

Vipul at 22:52, 5 April 2008

2008-04-05T22:52:15Z

← Older revision		Revision as of 22:52, 5 April 2008
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	for any three vector fields <math>X,Y,Z</math>.		for any three vector fields <math>X,Y,Z</math>.

	Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[Levi-Civita connection]] arising from a [[Riemannian metric]].		Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[Levi-Civita connection]] arising from a [[Riemannian metric]] or [[pseudo-Riemannian metric]].

	==Proof==		==Proof==

Vipul at 11:15, 30 August 2007

2007-08-30T11:15:10Z

New page

==Statement==

Let <math>\nabla</math> be a [[torsion-free linear connection]]. The [[Riemann curvature tensor]] <math>R</math> of <math>\nabla</math> satisfies the following '''first Bianchi identity''' or '''algebraic Bianchi identity''':

<math>R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0</math>

for any three vector fields <math>X,Y,Z</math>.

Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the [[Levi-Civita connection]] arising from a [[Riemannian metric]].

==Proof==

===Using repeated simplication and the Jacobi identity===

Let us plug the definition of the Riemann curvature tensor:

<math>\nabla_X\nabla_YZ - \nabla_Y\nabla_XZ + \nabla_Y\nabla_ZX - \nabla_Z\nabla_YX + \nabla_Z\nabla_XY - \nabla_X\nabla_ZY - \nabla_{[X,Y]}Z - \nabla_{[Y,Z]}X - \nabla_{[Z,X]}Y</math>

This can be regrouped as:

<math>\nabla_X(\nabla_YZ - \nabla_ZY) + \nabla_Y(\nabla_ZX - \nabla_XZ) + \nabla_Z (\nabla_XY - \nabla_YX) - \nabla_{[X,Y]}Z - \nabla_{[Y,Z]}X - \nabla_{[Z,X]}Y</math>

Now, since <math>\nabla</math> is torsion-free, we have <math>\nabla_YZ - \nabla_ZY = [Y,Z]</math> and similar simplifications yield:

<math>\nabla_X[Y,Z] + \nabla_Y[Z,X] + \nabla_Z[X,Y] - \nabla_{[X,Y]}Z - \nabla_{[Y,Z]}X - \nabla_{[Z,X]}Y</math>

again using the fact that <math>\nabla</math> is torsion-free, this simplifies to:

<math>[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]]</math>

this becomes zero by the [[Jacobi identity]].

===Using the differential Bianchi identity===

{{fillin}}