# Difference between revisions of "Torsion is tensorial"

From Diffgeom

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===Symbolic statement=== | ===Symbolic statement=== | ||

− | Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>). | + | Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[fact about::linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>). |

− | Consider the [[torsion of a linear connection|torsion]] of <math>\nabla</math>, namely: | + | Consider the [[fact about::torsion of a linear connection|torsion]] of <math>\nabla</math>, namely: |

<math>\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math> | <math>\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math> | ||

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<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [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 [[fact about::tensorial map]] in both coordinates. |

==Facts used== | ==Facts used== |

## Revision as of 01:19, 24 July 2009

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.