On Singular Semi-Riemannian Manifolds

TL;DR

This paper develops a geometric framework for singular semi-Riemannian manifolds with degenerate metrics, introducing invariant contractions and covariant derivatives to define curvature and Einstein's equations smoothly.

Contribution

It introduces a canonical contraction and covariant derivative for degenerate metrics, enabling smooth curvature and Einstein tensor formulations on singular semi-Riemannian manifolds.

Findings

Defined an invariant contraction for degenerate metrics.

Constructed a smooth Riemann curvature tensor on semi-regular manifolds.

Formulated a densitized Einstein equation valid even with metric singularities.

Abstract

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. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.