Curvature is antisymmetric in last two variables

Statement

Suppose $M$ is a differential manifold and $g$ is a Riemannian metric or pseudo-Riemannian metric and $\mathrm{\nabla }$ is the Levi-Civita connection for $g$. Consider the Riemann curvature tensor $R$ of $\mathrm{\nabla }$. 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

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