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Torsion is tensorial

1,073 bytes added, 17:43, 6 January 2012
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===
We'll use the fact that tensoriality ''Given''': <math>f:M \to \R</math> is equivalent to <math>C^\infty</math>-linearity.function
'''To prove''': <math>\tau(\nabla)(fX,Y) = f\tau(\nabla)(X,Y)</math>
'''Proof''': We prove this by expanding everything start out on 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>
To prove Each step below is obtained from the equality with <math>f \tau(\nabla)(X,Y)</math>, we observe that it reduces to showing:previous one via some manipulation explained along side.
{| class="sortable" border="1"! Step no. !! Current status of left side !! Facts/properties used !! Specific rewrites|-| 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 - [fXX,Y])</math>|| factor out || --|-which is exactly what the corollary | 7 || <math>f\tau(\nabla)(X,Y)</math> || use definition of Leibniz rule above states.torsion || <math>\nabla_X Y - \nabla_Y X - [X,Y] \mapsto \tau(\nabla)(X,Y)</math>|}
===Tensoriality in the second coordinate===
''To prove'' <math>\tau(\nabla)(X,fY) = f \tau(\nabla)(X,Y)</math>
''Proof'': We prove this by expanding everything out on This is similar to tensoriality in the left side: <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 resultfirst coordinate.
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