Torsion is tensorial
This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
View other such statements
Consider the torsion of , namely:
Then, is a tensorial map in both coordinates. In other words, the value of at a point depends only on and does not depend on the values of the vectors fields at points other than .
More explicitly, for any point , defines a bilinear map:
Further, since in fact torsion is antisymmetric, this is an alternating bilinear map.
|Fact no.||Name||Statement with symbols|
|1||Any connection is -linear in its subscript argument||for any -function and vector field .|
|2||The Leibniz-like 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 :
To prove tensoriality in a variable, it suffices to show -linearity in that variable. This is because linearity in -functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.
The proofs for and are analogous, and rely on manipulation of the Lie bracket and the property of a connection being in the subscript vector.
Tensoriality in the first coordinate
Given: is -function
Proof: We start out with the left side:
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||Definition of torsion||whole thing|
|2||Fact (1): -linearity of connection in subscript argument|
|3||Fact (2): The Leibniz-like axiom that's part of the definition of a connection|
|7||use definition of torsion|
Tensoriality in the second coordinate
The proof is analogous to that for the first coordinate.
Proof: This is similar to tensoriality in the first coordinate.