November 11, 2009
It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric has to satisfy certain conditions. (As usual, we denote by the Levi-Civita connection associated to , and we assume the ground manifold is smooth.)
First of all, we have skew-symmetry
This is immediate from the definition.
Next, we have another variant of skew-symmetry:
Proposition 1 (more…)
November 10, 2009
Ok, now onto the Levi-Civita connection. Fix a manifold with the pseudo-metric . This means essentially a metric, except that as a bilinear form on the tangent spaces is still symmetric and nondegenerate but not necessarily positive definite. It is still possible to say that a pseudo-metric is compatible with a given connection.
This is the fundamental theorem of Riemannian geometry:
Theorem 1 There is a unique symmetric connection on compatible with . (more…)