# Changes

## Curvature is tensorial

, 00:09, 20 December 2011
Proof
| 4 || $f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{(Yf)X + [fX,Y]}$ || $\nabla$ is additive in its subscript argument || $\nabla_{(Yf)X} + \nabla_{[fX,Y]} = \nabla_{(Yf)X + [fX,Y]}$
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| 5 || $f(\nabla_X\nabla_Y - \nabla_Y\nabla_X ) - \nabla_{f[X,Y]})$ || Fact (3) || $[fX,Y] + (Yf)X \to f[X,Y]$.|-| 6 || $f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}) || Fact (1) || [itex]\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}$
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| 4 || $f(\nabla_X\nabla_Y - \nabla_Y\nabla_X) - \nabla_{[X,fY] - (Xf)Y}$ || $\nabla$ is additive in its subscript argument. || $\nabla_{[X,fY]} - \nabla_{(Xf)Y} \to \nabla_{[X,fY] - (Xf)Y}$.
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| 5 || $f(\nabla_X\nabla_Y - \nabla_Y\nabla_X ) - \nabla_{f[X,Y]}$ || Fact (3) || $[X,fY] - (Xf)Y \to f[X,Y]$|-| 6 || $f(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})$ || Fact (1) || $\nabla_{f[X,Y]} \to f\nabla_{[X,Y]}$
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