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Torsion is tensorial

290 bytes added, 17:36, 6 January 2012
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 !! Statement with symbols|-| 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 || 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>\! Xnabla</math>, we have <math>\nabla_A(fgfB) = (XfAf)(gB) + f\nabla_A(XgB)</math>|-* | 3 || [[uses::Corollary of Leibniz rule for Lie bracket]](in turn follows from [[uses: This states that :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> * 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>|}
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