# Curvature is antisymmetric in last two variables

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## Statement

Suppose $M$ is a differential manifold and $g$ is a Riemannian metric or pseudo-Riemannian metric and $\nabla$ is the Levi-Civita connection for $g$. Consider the Riemann curvature tensor $R$ of $\nabla$. In other words, $R$ is the Riemann curvature tensor of the Levi-Civita connection for $g$. We can treat $R$ as a $(0,4)$-tensor:

$\! R(X,Y,Z,W) = g(R(X,Y)Z,W)$.

Then:

$\! R(X,Y,Z,W) = R(X,Y,W,Z)$.

## Facts used

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

$R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0$.

## Proof

Fill this in later