Torsion is tensorial
From Diffgeom
This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
View other such statements
Contents
Statement
Symbolic statement
Let be a differential manifold and be a linear connection on (viz., is a connection on the tangent bundle of ).
Consider the torsion of , namely:
given by:
Then, is a tensorial map in both coordinates.
Facts used
Fact no.  Name  Statement with symbols 

1  Any connection is linear in its subscript argument  for any function and vector field . 
2  The Leibnizlike axiom that is part of the definition of a connection  For a function and vector fields , and a connection , we have 
3  Corollary of Leibniz rule for Lie bracket (in turn follows from Leibniz rule for derivations  For a function and vector fields :

Proof
Tensoriality in the first coordinate
We'll use the fact that tensoriality is equivalent to linearity.
To prove:
Proof: We prove this by expanding everything out on the left side:
To prove the equality with , we observe that it reduces to showing:
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
Proof: We prove this by expanding everything out on the left side:
(the last step uses the corollary of Leibniz rule).
Canceling terms, yields the required result.