A Note on Distributional Semi-Riemannian Geometry

TL;DR

This paper explores semi-Riemannian geometry under low-regularity conditions, comparing linear distributional frameworks with nonlinear Colombeau approaches to understand their differences and applications.

Contribution

It provides a comparative analysis of linear and nonlinear distributional geometries in semi-Riemannian contexts, highlighting their conceptual distinctions and potential uses.

Findings

Clarifies the relationship between Schwartz's linear and Colombeau's nonlinear distributional geometries.

Identifies key differences in handling low-regularity metrics.

Provides insights into the applicability of each framework in geometric analysis.

Abstract

We discuss some basic concepts of semi-Riemannian geometry in low-regularity situations. In particular, we compare the settings of (linear) distributional geometry in the sense of L. Schwartz and nonlinear distributional geometry in the sense of J.F. Colombeau.