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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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.

More explicitly, for any point p \in M, \tau(\nabla) defines a bilinear map:

\! \tau(\nabla): T_p(M) \times T_p(M) \to T_p(M)

Further, since in fact torsion is antisymmetric, this is an alternating bilinear map.

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

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.