Curvature is tensorial - Revision history

Vipul: /* Facts used */

2012-01-06T17:36:17Z

Facts used

← Older revision		Revision as of 17:36, 6 January 2012
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	\| 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>		\| 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>
	\|-		\|-
	\| 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>:		\| 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>:
	<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>\! f[X,Y] = [X,fY] - (Xf)Y</math>		<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>\! f[X,Y] = [X,fY] - (Xf)Y</math>
	\|}		\|}

Vipul: /* Facts used */

2011-12-20T00:16:16Z

Facts used

← Older revision		Revision as of 00:16, 20 December 2011
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	\|-		\|-
	\| 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>:		\| 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>:
	<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>f[X,Y] = [X,fY] - (Xf)Y</math>		<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>\! f[X,Y] = [X,fY] - (Xf)Y</math>
	\|}		\|}

Vipul: /* Tensoriality in the first variable */

2011-12-20T00:15:19Z

Tensoriality in the first variable

← Older revision		Revision as of 00:15, 20 December 2011
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	\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{f[X,Y]}</math> \|\| Fact (3) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>		\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{f[X,Y]}</math> \|\| Fact (3) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>
	\|-		\|-
	\| 6 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}) \|\| Fact (1) \|\| <math>\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}</math>		\| 6 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (1) \|\| <math>\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}</math>
	\|}		\|}

Vipul: /* Proof */

2011-12-20T00:09:31Z

Proof

← Older revision		Revision as of 00:09, 20 December 2011
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	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>
	\|-		\|-
	\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (3) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>.		\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{f[X,Y]}</math> \|\| Fact (3) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>
			\|-
			\| 6 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}) \|\| Fact (1) \|\| <math>\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}</math>
	\|}		\|}

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	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.
	\|-		\|-
	\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math> \|\| Fact (3) \|\| <math>[X,fY] - (Xf)Y \to f[X,Y]</math>		\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{f[X,Y]}</math> \|\| Fact (3) \|\| <math>[X,fY] - (Xf)Y \to f[X,Y]</math>
			\|-
			\| 6 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (1) \|\| <math>\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}</math>
	\|}		\|}

Vipul: /* Proof */

2011-02-25T21:09:52Z

Proof

← Older revision		Revision as of 21:09, 25 February 2011
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	\| 1 \|\| <math>f\nabla_X\nabla_Y - \nabla_Y (f \nabla_X) - \nabla_{[fX,Y]}</math> \|\| Fact (1): <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>\nabla_{fX} \to f\nabla_X</math>		\| 1 \|\| <math>f\nabla_X\nabla_Y - \nabla_Y (f \nabla_X) - \nabla_{[fX,Y]}</math> \|\| Fact (1): <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>\nabla_{fX} \to f\nabla_X</math>
	\|-		\|-
	\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (2)~~, the Leibniz-like axiom for connection.~~ \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.		\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (2) \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.
	\|-		\|-
	\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| Fact (1)~~: <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument~~ \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>		\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| Fact (1) \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>
	\|-		\|-
	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>
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	! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites		! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
	\|-		\|-
	\| 1 \|\| <math>\nabla_X(f\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (1)~~: <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument~~ \|\| <math>\nabla_{fY} \to f\nabla_Y</math>.		\| 1 \|\| <math>\nabla_X(f\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (1) \|\| <math>\nabla_{fY} \to f\nabla_Y</math>.
	\|-		\|-
	\| 2 \|\| <math>(Xf)\nabla_Y + f(\nabla_X\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (2)~~, the Leibniz-like axiom for connection~~ \|\| <math>\nabla_X(f\nabla_Y) \to (Xf)\nabla_Y + f(\nabla_X\nabla_Y)</math>. To make this more concrete, imagine an input <math>Z</math>. Then, the rewrite becomes <math>\nabla_X(f\nabla_Y(Z)) \to (Xf)\nabla_Y(X) + f(\nabla_X\nabla_Y(Z))</math>. This comes setting <math>A = X</math>, <math>f = f</math>, <math>B = \nabla_YZ</math> in Fact (3).		\| 2 \|\| <math>(Xf)\nabla_Y + f(\nabla_X\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (2) \|\| <math>\nabla_X(f\nabla_Y) \to (Xf)\nabla_Y + f(\nabla_X\nabla_Y)</math>. To make this more concrete, imagine an input <math>Z</math>. Then, the rewrite becomes <math>\nabla_X(f\nabla_Y(Z)) \to (Xf)\nabla_Y(X) + f(\nabla_X\nabla_Y(Z))</math>. This comes setting <math>A = X</math>, <math>f = f</math>, <math>B = \nabla_YZ</math> in Fact (3).
	\|-		\|-
	\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]} + \nabla_{(Xf)Y}</math> \|\| ~~<math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument.~~ \|\| <math>(Xf)\nabla_Y \to \nabla_{(Xf)Y}</math>		\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]} + \nabla_{(Xf)Y}</math> \|\| Fact (1) \|\| <math>(Xf)\nabla_Y \to \nabla_{(Xf)Y}</math>
	\|-		\|-
	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.
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	===Tensoriality in the third variable===		===Tensoriality in the third variable===

	~~Let~~ <math>f: M \to \R</math> ~~be a scalar function~~. ~~We will show that:~~		'''Given''': A <math>C^\infty</math>-function <math>f:M \to \R</math>.

	<math>\! R(X,Y) (fZ) = f R(X,Y) Z</math>		'''To prove''': <math>\! R(X,Y) (fZ) = f R(X,Y) Z</math>. More explicitly, <math>\! \nabla_X\nabla_Y(fZ) - \nabla_Y\nabla_X(fZ) - \nabla_{[X,Y]}(fZ) = f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z</math>.

	We start out with the left side:		We start out with the left side:
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	<math>\nabla_X\nabla_Y(fZ) - \nabla_Y\nabla_X(fZ) - \nabla_{[X,Y]}(fZ)</math>		<math>\nabla_X\nabla_Y(fZ) - \nabla_Y\nabla_X(fZ) - \nabla_{[X,Y]}(fZ)</math>

	~~Now we apply~~ the ~~Leibniz rule for connnections on each term:~~		Each step below is obtained from the previous one via some manipulation explained along side.

	<math>\nabla_X( (Yf)(Z) + f \nabla_YZ) - \nabla_Y ((Xf)Z + f \nabla_XZ) - f \nabla_{[X,Y]}Z - ([X,Y]f) Z</math>		{\| class="sortable" border="1"
			! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
	~~We again apply the Leibniz rule~~ to ~~the first two term groups:~~		\|-
			\| 1 \|\| <math>\! \nabla_X( (Yf)(Z) + f \nabla_YZ) - \nabla_Y ((Xf)Z + f \nabla_XZ) - f \nabla_{[X,Y]}Z - ([X,Y]f) Z</math> \|\| Fact (2) \|\| <math>\nabla_Y(fZ) \to (Yf)(Z) + f\nabla_YZ</math> and <math>\nabla_X(fZ) \to (Xf)Z + f\nabla_XZ</math>
	<math>(XYf)(Z) + (Yf) \nabla_XZ + (Xf) \nabla_YZ + f \nabla_X\nabla_YZ - (YXf)Z - (Xf) \nabla_YZ - (Yf) \nabla_XZ -f \nabla_Y\nabla_XZ - f \nabla_{[X,Y]}Z - ([X,Y]f) Z</math>		\|-
			\| 2 \|\| <math>\! (XYf)(Z) + (Yf) \nabla_XZ + (Xf) \nabla_YZ + f \nabla_X\nabla_YZ - (YXf)Z - (Xf) \nabla_YZ - (Yf) \nabla_XZ -f \nabla_Y\nabla_XZ - f \nabla_{[X,Y]}Z - ([X,Y]f) Z</math> \|\| Fact (2) \|\| <math>\nabla_X((Yf)Z) \to X((Yf)Z) + (Yf)\nabla_XZ</math>, etc.
	~~After cancellations we are left with the following six terms:~~		\|-
			\| 3 \|\| <math>f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z</math> \|\| -- \|\| cancellations
	<math>f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z</math>		\|-
			\| 4 \|\| <math>f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z + ((XY - YX - [X,Y])f)Z</math>\|\| use <math>[X,Y] = XY - YX</math>, definition \|\| cancellation
	~~But since~~ <math>[X,Y] = XY - YX</math>~~, the last three terms vanish, and we are left with:~~		\|}

	<math>~~\! f R(~~X,Y)Z</math>

Vipul at 21:03, 25 February 2011

2011-02-25T21:03:55Z

← Older revision		Revision as of 21:03, 25 February 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Fact no. !! Name !! ~~Full statement~~		! Fact no. !! Name !! Statement with symbols
	\|-		\|-
	\| 1 \|\| ~~[[uses::Leibniz rule for derivations]]~~\|\| ~~For a vector field~~ <math>X</math> ~~and functions~~ <math>f,g</math>~~, we have:~~ <math>~~\! X(fg) = (Xf)(g) + f(Xg)~~</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>.
	\|-		\|-
	\| 2 \|\| [[uses::Corollary of Leibniz rule for Lie bracket]]\|\| For a function <math>f</math> and vector fields <math>X,Y</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>
			\|-
			\| 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>:
	<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>f[X,Y] = [X,fY] - (Xf)Y</math>		<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>f[X,Y] = [X,fY] - (Xf)Y</math>
	\|-
	\| 3 \|\| 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>
	\|}		\|}

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	! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites		! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
	\|-		\|-
	\| 1 \|\| <math>f\nabla_X\nabla_Y - \nabla_Y (f \nabla_X) - \nabla_{[fX,Y]}</math> \|\| ~~By definition of a [[connection]],~~ <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>\nabla_{fX} \to f\nabla_X</math>		\| 1 \|\| <math>f\nabla_X\nabla_Y - \nabla_Y (f \nabla_X) - \nabla_{[fX,Y]}</math> \|\| Fact (1): <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>\nabla_{fX} \to f\nabla_X</math>
	\|-		\|-
	\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (3), the Leibniz-like axiom for connection. \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.		\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (2), the Leibniz-like axiom for connection. \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.
	\|-		\|-
	\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>		\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| Fact (1): <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>
	\|-		\|-
	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>
	\|-		\|-
	\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (2) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>.		\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (3) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>.
	\|}		\|}

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	! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites		! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
	\|-		\|-
	\| 1 \|\| <math>\nabla_X(f\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| ~~By definition of a connection,~~ <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>\nabla_{fY} \to f\nabla_Y</math>.		\| 1 \|\| <math>\nabla_X(f\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (1): <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>\nabla_{fY} \to f\nabla_Y</math>.
	\|-		\|-
	\| 2 \|\| <math>(Xf)\nabla_Y + f(\nabla_X\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (3), the Leibniz-like axiom for connection \|\| <math>\nabla_X(f\nabla_Y) \to (Xf)\nabla_Y + f(\nabla_X\nabla_Y)</math>. To make this more concrete, imagine an input <math>Z</math>. Then, the rewrite becomes <math>\nabla_X(f\nabla_Y(Z)) \to (Xf)\nabla_Y(X) + f(\nabla_X\nabla_Y(Z))</math>. This comes setting <math>A = X</math>, <math>f = f</math>, <math>B = \nabla_YZ</math> in Fact (3).		\| 2 \|\| <math>(Xf)\nabla_Y + f(\nabla_X\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (2), the Leibniz-like axiom for connection \|\| <math>\nabla_X(f\nabla_Y) \to (Xf)\nabla_Y + f(\nabla_X\nabla_Y)</math>. To make this more concrete, imagine an input <math>Z</math>. Then, the rewrite becomes <math>\nabla_X(f\nabla_Y(Z)) \to (Xf)\nabla_Y(X) + f(\nabla_X\nabla_Y(Z))</math>. This comes setting <math>A = X</math>, <math>f = f</math>, <math>B = \nabla_YZ</math> in Fact (3).
	\|-		\|-
	\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]} + \nabla_{(Xf)Y}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>(Xf)\nabla_Y \to \nabla_{(Xf)Y}</math>		\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]} + \nabla_{(Xf)Y}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>(Xf)\nabla_Y \to \nabla_{(Xf)Y}</math>
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	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.
	\|-		\|-
	\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math> \|\| Fact (2) \|\| <math>[X,fY] - (Xf)Y \to f[X,Y]</math>		\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math> \|\| Fact (3) \|\| <math>[X,fY] - (Xf)Y \to f[X,Y]</math>
	\|}		\|}

Vipul: /* Tensoriality in the first variable */

2011-02-25T21:01:02Z

Tensoriality in the first variable

← Older revision		Revision as of 21:01, 25 February 2011
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	\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>		\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>
	\|-		\|-
	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)}X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>
	\|-		\|-
	\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (2) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>.		\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (2) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>.

Vipul: /* Proof */

2011-02-25T21:00:10Z

Proof

← Older revision		Revision as of 21:00, 25 February 2011
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	\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (3), the Leibniz-like axiom for connection. \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.		\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (3), the Leibniz-like axiom for connection. \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.
	\|-		\|-
	\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <~~nath~~>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>		\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X} - \nabla_{[fX,Y]}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>
	\|-		\|-
	\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (2) \|\| <math>[fX,Y] \to f[X,Y] ~~- (Yf)X~~</math>.		\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is additive in its subscript argument \|\| <math>\nabla_{(Yf)}X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}</math>
			\|-
			\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (2) \|\| <math>[fX,Y] + (Yf)X \to f[X,Y]</math>.
	\|}		\|}

	===Tensoriality in the second variable===		===Tensoriality in the second variable===

	~~Let~~ <math>f:M \to \R</math> be a ~~scalar~~ function. ~~We will show that:~~		'''Given''': <math>f:M \to \R</math> is a <math>C^\infty</math>-function.

	<math>\! R(X,fY) = f R(X,Y)</math>		'''To prove''': <math>\! R(X,fY) = f R(X,Y)</math>, or more explicitly, <math>\nabla_X\nabla_{fY} - \nabla_{fY}\nabla_X - \nabla_{[X,fY]} = f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math>.

	We start out with the left side:		We start out with the left side:
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	<math>\nabla_X\nabla_{fY} - \nabla_{fY}\nabla_X - \nabla_{[X,fY]}</math>		<math>\nabla_X\nabla_{fY} - \nabla_{fY}\nabla_X - \nabla_{[X,fY]}</math>

	~~Applying~~ the ~~Leibniz rule and the property of a connection being <math>C^\infty</math> in its subscript variable yields:~~		Each step below is obtained from the previous one via some manipulation explained along side.

	<math>(Xf)\nabla_Y + f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]}</math>		{\| class="sortable" border="1"
			! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
	~~which simplifies~~ to:		\|-
			\| 1 \|\| <math>\nabla_X(f\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| By definition of a connection, <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <math>\nabla_{fY} \to f\nabla_Y</math>.
	<math>f(\nabla_X\~~nabla_y~~ - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math>		\|-
			\| 2 \|\| <math>(Xf)\nabla_Y + f(\nabla_X\nabla_Y) - f\nabla_Y\nabla_X - \nabla_{[X,fY]}</math> \|\| Fact (3), the Leibniz-like axiom for connection \|\| <math>\nabla_X(f\nabla_Y) \to (Xf)\nabla_Y + f(\nabla_X\nabla_Y)</math>. To make this more concrete, imagine an input <math>Z</math>. Then, the rewrite becomes <math>\nabla_X(f\nabla_Y(Z)) \to (Xf)\nabla_Y(X) + f(\nabla_X\nabla_Y(Z))</math>. This comes setting <math>A = X</math>, <math>f = f</math>, <math>B = \nabla_YZ</math> in Fact (3).
	~~We now use the corollary stated above:~~		\|-
			\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY]} + \nabla_{(Xf)Y}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>(Xf)\nabla_Y \to \nabla_{(Xf)Y}</math>
	<math>\~~! f~~[X,Y] = [X,fY] - (Xf)Y</math>		\|-
			\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}</math> \|\| <math>\nabla</math> is additive in its subscript argument. \|\| <math>\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}</math>.
	~~substituting this gives:~~		\|-
			\| 5 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math> \|\| Fact (2) \|\| <math>[X,fY] - (Xf)Y \to f[X,Y]</math>
	<math>f (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math>		\|}

	~~which is~~ <math>f R(X,Y)</math>

	===Tensoriality in the third variable===		===Tensoriality in the third variable===

Vipul at 20:45, 25 February 2011

2011-02-25T20:45:06Z

← Older revision		Revision as of 20:45, 25 February 2011
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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:		{\| class="sortable" border="1"
			! Fact no. !! Name !! Full statement
	<math>X(fg) = (Xf)(g) + f(Xg)</math>		\|-
			\| 1 \|\| [[uses::Leibniz rule for derivations]]\|\| For a vector field <math>X</math> and functions <math>f,g</math>, we have: <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>:		\|-
			\| 2 \|\| [[uses::Corollary of Leibniz rule for Lie bracket]]\|\| For a function <math>f</math> and vector fields <math>X,Y</math>:
	<math>f[X,Y] = [fX,Y] + (Yf)X</math>		<br><math>\! f[X,Y] = [fX,Y] + (Yf)X</math><br><math>f[X,Y] = [X,fY] - (Xf)Y</math>
			\|-
	<math>f[X,Y] = [X,fY] - (Xf)Y</math>		\| 3 \|\| 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>
			\|}
	* 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>

	==Proof==		==Proof==
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	===Tensoriality in the first variable===		===Tensoriality in the first variable===

	~~Let~~ <math>f:M \to \R</math> be a ~~scalar~~ function. ~~We will show that:~~		'''Given''': <math>f:M \to \R</math> is a <math>C^\infty</math>-function.

	<math>\! R(fX,Y) = f R(X,Y)</math>		'''To prove''': <math>\! R(fX,Y) = f R(X,Y)</math>, or more explicitly, <math>\! \nabla_{fX}\nabla_Y - \nabla_Y \nabla_{fX} - \nabla_{[fX,Y]} = f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math>

	We start out with the left side:		We start out with the left side:
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	<math>\nabla_{fX}\nabla_Y - \nabla_Y \nabla_{fX} - \nabla_{[fX,Y]}</math>		<math>\nabla_{fX}\nabla_Y - \nabla_Y \nabla_{fX} - \nabla_{[fX,Y]}</math>

	~~Now by~~ the ~~definition of a [[connection]], <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument~~. ~~Thus, the above expression can be written as:~~		Each step below is obtained from the previous one via some manipulation explained along side.

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

	~~Now applying the Leibniz rule for connections, we get:~~

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

	~~We can rewrite <math>(Yf)\nabla_X = \nabla_{(Yf)X}</math> and we then get:~~

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

	~~By the corollary stated above, we have:~~

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

	~~which, substituted back, gives:~~

	<math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math>		{\| class="sortable" border="1"
			! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites
			\|-
			\| 1 \|\| <math>f\nabla_X\nabla_Y - \nabla_Y (f \nabla_X) - \nabla_{[fX,Y]}</math> \|\| By definition of a [[connection]], <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument. \|\| <math>\nabla_{fX} \to f\nabla_X</math>
			\|-
			\| 2 \|\| <math>f\nabla_X\nabla_Y - (Yf)\nabla_X - f \nabla_Y\nabla_X - \nabla_{[fX,Y]}</math> \|\| Fact (3), the Leibniz-like axiom for connection. \|\| <math>\nabla_Y(f \nabla_X) \to (Yf)\nabla_X + f\nabla_Y\nabla_X</math>. To understand this more clearly imagine an input <math>Z</math> to the whole expression, so that the rewrite becomes <math>\nabla_Y(f \nabla_X(Z)) \to (Yf)\nabla_X(Z) + f\nabla_Y\nabla_X(Z)</math>. In the notation of fact (3), <math>A = Y</math>, <math>f = f</math>, and <math>B = \nabla_X(Z)</math>.
			\|-
			\| 3 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}</math> \|\| <math>\nabla</math> is <math>C^\infty</math>-linear in its subscript argument \|\| <nath>(Yf)\nabla_X \to \nabla_{(Yf)X}</math>
			\|-
			\| 4 \|\| <math>f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})</math> \|\| Fact (2) \|\| <math>[fX,Y] \to f[X,Y] - (Yf)X</math>.
			\|}

	===Tensoriality in the second variable===		===Tensoriality in the second variable===

Vipul: /* Related facts */

2009-07-24T01:54:03Z

Related facts

← Older revision		Revision as of 01:54, 24 July 2009
Line 11:		Line 11:

	* [[Curvature is antisymmetric in first two variables]]		* [[Curvature is antisymmetric in first two variables]]
			* [[Curvature is antisymmetric in last two variables]]
			* [[Curvature is symmetric in the pairs of first and last two variables]]

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