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Identities for the curvature tensor November 11, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric {g} has to satisfy certain conditions.  (As usual, we denote by \nabla the Levi-Civita connection associated to g, and we assume the ground manifold is smooth.)

First of all, we have skew-symmetry

\displaystyle R(X,Y)Z = -R(Y,X)Z. 

This is immediate from the definition.

Next, we have another variant of skew-symmetry:

Proposition 1 \displaystyle g( R(X,Y) Z, W) = -g( R(X,Y) W, Z)  (more…)