Curvature is antisymmetric in last two variables
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:
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:
Substituting (1) and (2) in yields .