# First Bianchi identity

From Diffgeom

## Contents

## Statement

Let be a torsion-free linear connection. The Riemann curvature tensor of satisfies the following **first Bianchi identity** or **algebraic Bianchi identity**:

for any three vector fields .

Notice that since this proof is applicable for any torsion-free linear connection, it in particular holds for the Levi-Civita connection arising from a Riemannian metric or pseudo-Riemannian metric.

## Related facts

- Second Bianchi identity (also called the
*differential Bianchi identity*). - Curvature is tensorial
- Torsion is tensorial

## Proof

### Using repeated simplication and the Jacobi identity

Let us plug the definition of the Riemann curvature tensor:

This can be regrouped as:

Now, since is torsion-free, we have and similar simplifications yield:

again using the fact that is torsion-free, this simplifies to:

this becomes zero by the Jacobi identity.

### Using the differential Bianchi identity

*Fill this in later*