# Difference between revisions of "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

## 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. In other words, the value of $\tau(\nabla)$ at a point $p \in M$ depends only on $\nabla, X(p), Y(p)$ and does not depend on the values of the vectors fields $X,Y$ at points other than $p$.

## 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

To prove tensoriality in a variable, it suffices to show $C^\infty$-linearity in that variable. This is because linearity in $C^\infty$-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 $C^\infty$ in the subscript vector.

### Tensoriality in the first coordinate

Given: $f:M \to \R$ is $C^\infty$-function

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

Proof: We start out with the left side: $\tau(\nabla)(fX,Y)$

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 $\nabla_{fX}(Y) - \nabla_Y(fX) - [fX,Y]$ Definition of torsion whole thing
2 $f \nabla_X Y - \nabla_Y(fX) - [fX,Y]$ Fact (1): $C^\infty$-linearity of connection in subscript argument $\nabla_{fX} \mapsto f\nabla_X$
3 $f \nabla_X Y - (f \nabla_Y X + (Yf)(X)) - [fX,Y]$ Fact (2): The Leibniz-like axiom that's part of the definition of a connection $\nabla_Y(fX) \mapsto f\nabla_YX + (Yf)(X)$
4 $f \nabla_X Y - f \nabla_Y X - ((Yf)(X) + [fX,Y])$ parenthesis rearrangement --
5 $f \nabla_X Y - f \nabla_Y X - f[X,Y]$ Fact (3) $(Yf)(X) + [fX,Y] \mapsto f[X,Y]$
6 $f(\nabla_X Y - \nabla_Y X - [X,Y])$ factor out --
7 $f\tau(\nabla)(X,Y)$ use definition of torsion $\nabla_X Y - \nabla_Y X - [X,Y] \mapsto \tau(\nabla)(X,Y)$

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