# Curvature is antisymmetric in last two variables

## Statement

Suppose is a differential manifold and is a Riemannian metric or pseudo-Riemannian metric and is the Levi-Civita connection for . Consider the Riemann curvature tensor of . In other words, is the Riemann curvature tensor of the Levi-Civita connection for . We can treat as a -tensor:

.

Then:

.

## Related facts

## Proof

We consider the expression :

By the bilinearity of , this simplifies to:

To prove that this is zero, it thus suffices to show that:

.

We now show . Since is a metric connection, the left side simplifies to:

.

Simplifying each of the two terms on the right side of , we get:

.

And:

.

Substituting (1) and (2) in yields .