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

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

Consider the torsion of $\mathrm{\nabla }$, namely:

$\tau (\mathrm{\nabla }):\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)$

given by:

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

Then, $\tau (\mathrm{\nabla })$ is a tensorial map in both coordinates. In other words, the value of $\tau (\mathrm{\nabla })$ at a point $p\in M$ depends only on $\mathrm{\nabla },X(p),Y(p)$ and does not depend on the values of the vectors fields $X,Y$ at points other than $p$.

More explicitly, for any point $p\in M$, $\tau (\mathrm{\nabla })$ defines a bilinear map:

$\tau (\mathrm{\nabla }):T_p(M)\times T_p(M)\to T_p(M)$

Further, since in fact torsion is antisymmetric, this is an alternating bilinear map.

Related facts

• Curvature is tensorial
• Torsion is antisymmetric
• Curvature is antisymmetric in first two variables

Facts used

Fact no.	Name	Statement with symbols
1	Any connection is $C^{\mathrm{\infty }}$-linear in its subscript argument	${\mathrm{\nabla }}_{fA}=f{\mathrm{\nabla }}_A$ for any $C^{\mathrm{\infty }}$-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 $\mathrm{\nabla }$, we have ${\mathrm{\nabla }}_A(fB)=(Af)(B)+f{\mathrm{\nabla }}_A(B)$
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

To prove tensoriality in a variable, it suffices to show $C^{\mathrm{\infty }}$-linearity in that variable. This is because linearity in $C^{\mathrm{\infty }}$-functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.

The proofs for $X$ and $Y$ are analogous, and rely on manipulation of the Lie bracket $[fX,Y]$ and the property of a connection being $C^{\mathrm{\infty }}$ in the subscript vector.

Tensoriality in the first coordinate

Given: $f:M\to \mathbb{R}$ is $C^{\mathrm{\infty }}$-function

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

Proof: We start out with the left side:

$\tau (\mathrm{\nabla })(fX,Y)$

Each step below is obtained from the previous one via some manipulation explained along side.

Step no.	Current status of left side	Facts/properties used	Specific rewrites
1	${\mathrm{\nabla }}_{fX}(Y)-{\mathrm{\nabla }}_Y(fX)-[fX,Y]$	Definition of torsion	whole thing
2	$f{\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_Y(fX)-[fX,Y]$	Fact (1): $C^{\mathrm{\infty }}$-linearity of connection in subscript argument	${\mathrm{\nabla }}_{fX}\mapsto f{\mathrm{\nabla }}_X$
3	$f{\mathrm{\nabla }}_{X}Y-(f{\mathrm{\nabla }}_{Y}X+(Yf)(X))-[fX,Y]$	Fact (2): The Leibniz-like axiom that's part of the definition of a connection	${\mathrm{\nabla }}_Y(fX)\mapsto f{\mathrm{\nabla }}_{Y}X+(Yf)(X)$
4	$f{\mathrm{\nabla }}_{X}Y-f{\mathrm{\nabla }}_{Y}X-((Yf)(X)+[fX,Y])$	parenthesis rearrangement	--
5	$f{\mathrm{\nabla }}_{X}Y-f{\mathrm{\nabla }}_{Y}X-f[X,Y]$	Fact (3)	$(Yf)(X)+[fX,Y]\mapsto f[X,Y]$
6	$f({\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y])$	factor out	--
7	$f\tau (\mathrm{\nabla })(X,Y)$	use definition of torsion	${\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y]\mapsto \tau (\mathrm{\nabla })(X,Y)$

Tensoriality in the second coordinate

The proof is analogous to that for the first coordinate.

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

Proof: This is similar to tensoriality in the first coordinate.