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

From Diffgeom

## Contents

## Statement

Suppose is a differential manifold and is a Riemannian metric or pseudo-Riemannian metric on . Suppose is the Levi-Civita connection on and is the Riemann curvature tensor for . (In other words, is the Riemann curvature tensor of the Levi-Civita connection).

We can view as a -tensor as follows:

.

Then, we have:

.

Note that since curvature is antisymmetric in first two variables and curvature is antisymmetric in last two variables, this essentially shows that is a symmetric bilinear form on .

## Related facts

- Curvature is tensorial
- Curvature is antisymmetric in first two variables
- Curvature is antisymmetric in last two variables
- First Bianchi identity
- Second Bianchi identity

## Facts used

- Curvature is antisymmetric in first two variables
- Curvature is antisymmetric in last two variables
- First Bianchi identity: This states that if is a torsion-free linear connection, and is its Riemann curvature tensor, then:

.

For the corresponding -tensor, we have:

.

## Proof

Applying fact (3), we get:

Similar statements, permuting the variables, are:

Consider (1) + (2) - (3) - (4) and uses facts (1) and (2). We get:

This completes the proof.