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Curvature is tensorial

281 bytes added, 00:09, 20 December 2011
Proof
| 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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| 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>
|}
| 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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| 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>
|}
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