Torsion is tensorial

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This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
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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.

Related facts

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


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:


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.