Difference between revisions of "Curvature is antisymmetric in last two variables"
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Revision as of 01:28, 24 July 2009
Statement
Suppose is a differential manifold and is a Riemannian metric or pseudoRiemannian metric and is the LeviCivita connection for . Consider the Riemann curvature tensor of . In other words, is the Riemann curvature tensor of the LeviCivita connection for . We can treat as a tensor:
.
Then:
.
Facts used
 First Bianchi identity: This states that if is a torsionfree linear connection, then:
.
Proof
Fill this in later