# Difference between revisions of "Curvature is tensorial"

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* [[Curvature is antisymmetric in first two variables]] | * [[Curvature is antisymmetric in first two variables]] | ||

+ | * [[Curvature is antisymmetric in last two variables]] | ||

+ | * [[Curvature is symmetric in the pairs of first and last two variables]] | ||

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

## Revision as of 01:54, 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

Let be a connection on a vector bundle over a differential manifold . The Riemann curvature tensor of is given as a map defined by:

We claim that is a tensorial map in each of the variables .

## Related facts

- Curvature is antisymmetric in first two variables
- Curvature is antisymmetric in last two variables
- Curvature is symmetric in the pairs of first and last two variables

## 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

To prove tensoriality in a variable, it suffices to show -linearity in that variable. This is because linearity in -functions guarantees linearity in a function that is 1 at exactly one point, and zero at others.

The proofs for and are analogous, and rely on manipulation of the Lie bracket and the property of a connection being in the subscript vector. These proofs do not involve any explicit use of . The proof for relies simply on repeated application of the product rule, and the fact that .

### Tensoriality in the first variable

Let be a scalar function. We will show that:

We start out with the left side:

Now by the definition of a connection, is -linear in its subscript argument. Thus, the above expression can be written as:

Now applying the Leibniz rule for connections, we get:

We can rewrite and we then get:

By the corollary stated above, we have:

which, substituted back, gives:

### Tensoriality in the second variable

Let be a scalar function. We will show that:

We start out with the left side:

Applying the Leibniz rule and the property of a connection being in its subscript variable yields:

which simplifies to:

We now use the corollary stated above:

substituting this gives:

which is

### Tensoriality in the third variable

Let be a scalar function. We will show that:

We start out with the left side:

Now we apply the Leibniz rule for connnections on each term:

We again apply the Leibniz rule to the first two term groups:

After cancellations we are left with the following six terms:

But since , the last three terms vanish, and we are left with: