Torsion is tensorial: Difference between revisions

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given by:
given by:


<math>\tau(\nabla)(X,Y) = \nabla_Y X - \nabla_X Y - [X,Y]</math>
<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math>


Then, <math>\tau(\nabla)</math> is a [[tensorial map]] in both coordinates.
Then, <math>\tau(\nabla)</math> is a [[tensorial map]] in both coordinates.

Revision as of 22:58, 5 April 2008

This article gives the statement, and possibly proof, that a map constructed in a certain way is tensorial
View other such statements

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

  • 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:

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.

Fill this in later