# Curvature is antisymmetric in last two variables

From Diffgeom

## 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:

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Then:

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## Related facts

- Curvature is tensorial
- Curvature is antisymmetric in first two variables
- Curvature is symmetric in the pairs of first and last two variables

## Proof

We consider the expression :

By the bilinearity of , this simplifies to:

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

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

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

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And:

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Substituting (1) and (2) in yields .