On Singular Semi-Riemannian Manifolds

Understand

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature.

• If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices.
• In this article we develop the geometry of singular semi-Riemannian manifolds.
• First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics.