# Curvature is antisymmetric in last two variables

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## 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:

.

## Facts used

- First Bianchi identity: This states that if is a torsion-free linear connection, then:

.

## Proof

*Fill this in later*