# 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

- Leibniz rule for derivations: This states that for a vector field and functions , we have:

- Corollary of Leibniz rule for Lie bracket: This states that for a function and vector fields :

- The Leibniz rule axiom that's part of the definition of a connection, namely:

## 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.