# Spacetimes with distributional semi-Riemannian metrics and their curvature

## TL;DR

This paper develops a geometric framework for generalized sections of vector bundles, enabling the analysis of singular metrics like those describing cosmic strings, and extends classical differential geometry operations to these generalized contexts.

## Contribution

It introduces a comprehensive, localizable framework for nonlinear generalized sections that includes distributional sections and extends classical geometric operations.

## Key findings

- Calculated curvature of conical metric for cosmic strings
- Extended differential geometric operations to generalized sections
- Provided a localizable framework compatible with distributional calculus

## Abstract

We develop a comprehensive geometric framework for defining spaces $\mathcal{G}(M,E)$ of nonlinear generalized sections of vector bundles $E \to M$ containing spaces of distributional sections $\mathcal{D}'(M, E)$. Our theory incorporates classical differential geometric operations (like tensor products, covariant derivatives and Lie derivatives), is localizable and fully compatible with smooth and distributional tensor calculus. As an application to the treatment of singular metrics, we calculate the curvature of the conical metric used to describe cosmic strings.