# Topics in Lorentz Geometry

## TL;DR

This lecture note overview covers Lorentz geometry, including linear algebra in pseudo-Euclidean spaces, the fundamental theorem of curves in Lorentz-Minkowski space, and classification of timelike surfaces with constant curvature.

## Contribution

It provides a comprehensive synthesis of Lorentz geometry topics, including new adaptations for degenerate curves and surface classification methods.

## Key findings

- Extended fundamental theorem of curves to degenerate cases
- Classification of timelike surfaces with constant Gaussian curvature
- Application of split-complex algebra in Lorentz geometry

## Abstract

Lecture notes from the mini-course "Topics in Lorentz Geometry" taught at the University of S\~{a}o Paulo, in March/2019. The text has three parts: (i) an overall view of linear algebra in the pseudo-Euclidean space $\mathbb{R}^n_\nu$, with focus on Lorentz-Minkowski space and its role in Physics; (ii) a version of the Fundamental Theorem of Curves in 3-dimensional Lorentz-Minkowski space, with adaptations to curves with degenerate osculating plane; (iii) the problem of the diagonalization of the Weingarten map for timelike surfaces, local classification of surfaces with constant Gaussian curvature $K$ and Weierstrass' representation formula in Lorentz-Minkowski space (using split-complex algebra). V2: added references.