Torsion is tensorial - Revision history

Vipul: /* Statement */

2012-01-06T17:56:40Z

Statement

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	Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.		Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.

			More explicitly, for any point <math>p \in M</math>, <math>\tau(\nabla)</math> defines a bilinear map:

			<math>\! \tau(\nabla): T_p(M) \times T_p(M) \to T_p(M)</math>

			Further, since in fact [[torsion is antisymmetric]], this is an alternating bilinear map.

	==Related facts==		==Related facts==

Vipul: /* Statement */

2012-01-06T17:53:20Z

Statement

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	Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.		Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.

			==Related facts==

			* [[Curvature is tensorial]]
			* [[Torsion is antisymmetric]]
			* [[Curvature is antisymmetric in first two variables]]

	==Facts used==		==Facts used==

Vipul: /* Statement */

2012-01-06T17:52:23Z

Statement

← Older revision		Revision as of 17:52, 6 January 2012
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	<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math>		<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math>

	Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates.		Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates. In other words, the value of <math>\tau(\nabla)</math> at a point <math>p \in M</math> depends only on <math>\nabla, X(p), Y(p)</math> and does not depend on the values of the vectors fields <math>X,Y</math> at points other than <math>p</math>.

	==Facts used==		==Facts used==

Vipul: /* Statement */

2012-01-06T17:49:19Z

Statement

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	{{tensoriality fact}}		{{tensoriality fact}}
	==Statement==		==Statement==

	~~===Symbolic statement===~~

	Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[fact about::linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>).		Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[fact about::linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>).

Vipul: /* Proof */

2012-01-06T17:43:46Z

Proof

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	==Proof==		==Proof==

			To prove tensoriality in a variable, it suffices to show <math>C^\infty</math>-linearity in that variable. This is because linearity in <math>C^\infty</math>-functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.

			The proofs for <math>X</math> and <math>Y</math> are analogous, and rely on manipulation of the Lie bracket <math>[fX,Y]</math> and the property of a connection being <math>C^\infty</math> in the subscript vector.
	===Tensoriality in the first coordinate===		===Tensoriality in the first coordinate===

	We'~~ll use the fact that tensoriality~~ is ~~equivalent to~~ <math>C^\infty</math>-~~linearity.~~		'''Given''': <math>f:M \to \R</math> is <math>C^\infty</math>-function

	''To prove'': <math>\tau(\nabla)(fX,Y) = f\tau(\nabla)(X,Y)</math>		'''To prove''': <math>\tau(\nabla)(fX,Y) = f\tau(\nabla)(X,Y)</math>

	''Proof'': We ~~prove this by expanding everything~~ out on the left side:		'''Proof''': We start out with the left side:

	<math>\tau(\nabla)(fX,Y) ~~= \nabla_{fX}(Y) - \nabla_Y(fX) - [fX,Y] = f \nabla_X Y - f \nabla_Y X - (Yf)(X) - [fX,Y]~~</math>		<math>\tau(\nabla)(fX,Y)</math>

	~~To prove~~ the ~~equality with <math>f \tau(\nabla)(X,Y)</math>, we observe that it reduces to showing:~~		Each step below is obtained from the previous one via some manipulation explained along side.

	<math>\! (Yf)(X) = f[X,Y] - [fX,Y]</math>		{\| class="sortable" border="1"
			! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
	~~which is exactly what the corollary~~ of ~~Leibniz rule above states.~~		\|-
			\| 1 \|\| <math>\nabla_{fX}(Y) - \nabla_Y(fX) - [fX,Y]</math> \|\| Definition of torsion \|\| whole thing
			\|-
			\| 2 \|\| <math>f \nabla_X Y - \nabla_Y(fX) - [fX,Y]</math> \|\| Fact (1): <math>C^\infty</math>-linearity of connection in subscript argument \|\| <math>\nabla_{fX} \mapsto f\nabla_X</math>
			\|-
			\| 3 \|\| <math>f \nabla_X Y - (f \nabla_Y X + (Yf)(X)) - [fX,Y]</math> \|\| Fact (2): The Leibniz-like axiom that's part of the definition of a connection \|\| <math>\nabla_Y(fX) \mapsto f\nabla_YX + (Yf)(X)</math>
			\|-
			\| 4 \|\| <math>f \nabla_X Y - f \nabla_Y X - ((Yf)(X) + [fX,Y])</math> \|\| parenthesis rearrangement \|\| --
			\|-
			\| 5 \|\| <math>f \nabla_X Y - f \nabla_Y X - f[X,Y]</math> \|\| Fact (3) \|\| <math>(Yf)(X) + [fX,Y] \mapsto f[X,Y]</math>
			\|-
			\| 6 \|\| <math>f(\nabla_X Y - \nabla_Y X - [X,Y])</math> \|\| factor out \|\| --
			\|-
			\| 7 \|\| <math>f\tau(\nabla)(X,Y)</math> \|\| use definition of torsion \|\| <math>\nabla_X Y - \nabla_Y X - [X,Y] \mapsto \tau(\nabla)(X,Y)</math>
			\|}

	===Tensoriality in the second coordinate===		===Tensoriality in the second coordinate===
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	''To prove'' <math>\tau(\nabla)(X,fY) = f \tau(\nabla)(X,Y)</math>		''To prove'' <math>\tau(\nabla)(X,fY) = f \tau(\nabla)(X,Y)</math>

	''Proof'': ~~We prove this by expanding everything out on~~ the ~~left side:~~		''Proof'': This is similar to tensoriality in the first coordinate.

	~~<math>\tau(\nabla)(X,fY) = \nabla_X(fY) = \nabla_{fY}(X) - [X,fY] = (Xf)(Y) + f \nabla_XY - f\nabla_YX - f[X,Y] - (Xf)Y</math>~~

	~~(the last step uses the corollary of Leibniz rule).~~

	~~Canceling terms, yields the required result~~.

Vipul: /* Facts used */

2012-01-06T17:36:23Z

Facts used

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	==Facts used==		==Facts used==
	* [[Leibniz rule for derivations]]: This states that for a vector field <math>X</math> and functions <math>f,g</math>, we have:

	<math>\~~! X~~(fg) = (Xf)(g) + f(Xg)</math>		{\| class="sortable" border="1"
			! Fact no. !! Name !! Statement with symbols
	* [[Corollary of Leibniz rule for Lie bracket]]: ~~This states that~~ for a function <math>f</math> and vector fields <math>X,Y</math>:		\|-
			\| 1 \|\| Any connection is <math>C^\infty</math>-linear in its subscript argument \|\| <math>\nabla_{fA} = f\nabla_A</math> for any <math>C^\infty</math>-function <math>f</math> and vector field <math>A</math>.
	<math>\! f[X,Y] = [fX,Y] + (Yf)X</math>		\|-
			\| 2 \|\| The Leibniz-like axiom that is part of the definition of a connection \|\| For a function <math>f</math> and vector fields <math>A,B</math>, and a connection <math>\nabla</math>, we have <math>\nabla_A(fB) = (Af)(B) + f\nabla_A(B)</math>
	<math>\! f[X,Y] = [X,fY] - (Xf)Y</math>		\|-
			\| 3 \|\| [[uses::Corollary of Leibniz rule for Lie bracket]] (in turn follows from [[uses::Leibniz rule for derivations]]\|\| For a function <math>f</math> and vector fields <math>X,Y</math>:
	* The Leibniz rule axiom that's part of the definition of a [[connection]], namely:		<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>\! f[X,Y] = [X,fY] - (Xf)Y</math>
			\|}
	~~<math>\! \nabla_X(fZ) = (Xf)(Z) + f\nabla_X(Z)</math>~~

	==Proof==		==Proof==

Vipul at 01:19, 24 July 2009

2009-07-24T01:19:28Z

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	===Symbolic statement===		===Symbolic statement===

	Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>).		Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[fact about::linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>).

	Consider the [[torsion of a linear connection\|torsion]] of <math>\nabla</math>, namely:		Consider the [[fact about::torsion of a linear connection\|torsion]] of <math>\nabla</math>, namely:

	<math>\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>		<math>\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>
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	<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math>		<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math>

	Then, <math>\tau(\nabla)</math> is a [[tensorial map]] in both coordinates.		Then, <math>\tau(\nabla)</math> is a [[fact about::tensorial map]] in both coordinates.

	==Facts used==		==Facts used==

Vipul: /* Proof */

2009-07-24T01:18:57Z

Proof

← Older revision		Revision as of 01:18, 24 July 2009
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	To prove the equality with <math>f \tau(\nabla)(X,Y)</math>, we observe that it reduces to showing:		To prove the equality with <math>f \tau(\nabla)(X,Y)</math>, we observe that it reduces to showing:

	<math>(Yf)(X) = f[X,Y] - [fX,Y]</math>		<math>\! (Yf)(X) = f[X,Y] - [fX,Y]</math>

	which is exactly what the corollary of Leibniz rule above states.		which is exactly what the corollary of Leibniz rule above states.

Vipul: /* Facts used */

2009-07-24T01:18:22Z

Facts used

← Older revision		Revision as of 01:18, 24 July 2009
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	* [[Leibniz rule for derivations]]: This states that for a vector field <math>X</math> and functions <math>f,g</math>, we have:		* [[Leibniz rule for derivations]]: This states that for a vector field <math>X</math> and functions <math>f,g</math>, we have:

	<math>X(fg) = (Xf)(g) + f(Xg)</math>		<math>\! X(fg) = (Xf)(g) + f(Xg)</math>

	* [[Corollary of Leibniz rule for Lie bracket]]: This states that for a function <math>f</math> and vector fields <math>X,Y</math>:		* [[Corollary of Leibniz rule for Lie bracket]]: This states that for a function <math>f</math> and vector fields <math>X,Y</math>:

	<math>f[X,Y] = [fX,Y] + (Yf)X</math>		<math>\! f[X,Y] = [fX,Y] + (Yf)X</math>

	<math>f[X,Y] = [X,fY] - (Xf)Y</math>		<math>\! f[X,Y] = [X,fY] - (Xf)Y</math>

	* The Leibniz rule axiom that's part of the definition of a [[connection]], namely:		* The Leibniz rule axiom that's part of the definition of a [[connection]], namely:

	<math>\nabla_X(fZ) = (Xf)(Z) + f\nabla_X(Z)</math>		<math>\! \nabla_X(fZ) = (Xf)(Z) + f\nabla_X(Z)</math>

	==Proof==		==Proof==

Vipul: 10 revisions

2008-05-18T20:12:00Z

10 revisions

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