# Lorentzian surfaces and the curvature of the Schmidt metric

## TL;DR

This paper derives the general form of the Schmidt metric for Lorentzian surfaces, relates its Ricci scalar to that of the manifold, and explores applications in general relativity.

## Contribution

It provides the explicit form of the Schmidt metric for Lorentzian surfaces and links its curvature to the manifold's Ricci scalar, with practical implications.

## Key findings

- Derived the general form of the Schmidt metric for Lorentzian surfaces
- Expressed the Ricci scalar of the Schmidt metric in terms of the manifold's Ricci scalar
- Discussed applications to the theory of spacetime boundaries in general relativity

## Abstract

The b-boundary is a mathematical tool used to attach a topological boundary to incomplete Lorentzian manifolds using a Riemaniann metric called the Schmidt metric on the frame bundle. In this paper, we give the general form of the Schmidt metric in the case of Lorentzian surfaces. Furthermore, we write the Ricci scalar of the Schmidt metric in terms of the Ricci scalar of the Lorentzian manifold and give some examples. Finally, we discuss some applications to general relativity.