# Changes

## Torsion is tensorial

, 17:36, 6 January 2012
Facts used
==Facts used==
* [[Leibniz rule for derivations]]: This states that for a vector field $X$ and functions $f,g$, we have:
{| class="sortable" border="1"! Fact no. !! Name !! Statement with symbols|-| 1 || Any connection is $C^\infty$-linear in its subscript argument || $\nabla_{fA} = f\nabla_A$ for any $C^\infty$-function $f$ and vector field $A$.|-| 2 || The Leibniz-like axiom that is part of the definition of a connection || For a function $f$ and vector fields $A,B$, and a connection $\! Xnabla$, we have $\nabla_A(fgfB) = (XfAf)(gB) + f\nabla_A(XgB)$|-* | 3 || [[uses::Corollary of Leibniz rule for Lie bracket]](in turn follows from [[uses: This states that :Leibniz rule for derivations]]|| For a function $f$ and vector fields $X,Y$: <br>$\! f[X,Y] = [fX,Y] + (Yf)X$ <br>$\! f[X,Y] = [X,fY] - (Xf)Y$ * The Leibniz rule axiom that's part of the definition of a [[connection]], namely: $\! \nabla_X(fZ) = (Xf)(Z) + f\nabla_X(Z)$|}
==Proof==