Torsion is tensorial

This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
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Statement

Symbolic statement

Let $M$ be a differential manifold and $\nabla$ be a linear connection on $M$ (viz., $\nabla$ is a connection on the tangent bundle $TM$ of $M$).

Consider the torsion of $\nabla$, namely:

$\tau (\nabla ):\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$

given by:

$\tau (\nabla )(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$

Then, $\tau (\nabla )$ is a tensorial map in both coordinates.

Facts used

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 $\nabla$, we have $\nabla _{A}(fB)=(Af)(B)+f\nabla _{A}(B)$
3	Corollary of Leibniz rule for Lie bracket (in turn follows from Leibniz rule for derivations	For a function $f$ and vector fields $X,Y$: $\!f[X,Y]=[fX,Y]+(Yf)X$ $\!f[X,Y]=[X,fY]-(Xf)Y$

Proof

Tensoriality in the first coordinate

We'll use the fact that tensoriality is equivalent to $C^{\infty }$-linearity.

To prove: $\tau (\nabla )(fX,Y)=f\tau (\nabla )(X,Y)$

Proof: We prove this by expanding everything out on the left side:

$\tau (\nabla )(fX,Y)=\nabla _{fX}(Y)-\nabla _{Y}(fX)-[fX,Y]=f\nabla _{X}Y-f\nabla _{Y}X-(Yf)(X)-[fX,Y]$

To prove the equality with $f\tau (\nabla )(X,Y)$, we observe that it reduces to showing:

$\!(Yf)(X)=f[X,Y]-[fX,Y]$

which is exactly what the corollary of Leibniz rule above states.

Tensoriality in the second coordinate

The proof is analogous to that for the first coordinate.

To prove $\tau (\nabla )(X,fY)=f\tau (\nabla )(X,Y)$

Proof: We prove this by expanding everything out on the left side:

$\tau (\nabla )(X,fY)=\nabla _{X}(fY)=\nabla _{fY}(X)-[X,fY]=(Xf)(Y)+f\nabla _{X}Y-f\nabla _{Y}X-f[X,Y]-(Xf)Y$

(the last step uses the corollary of Leibniz rule).

Canceling terms, yields the required result.