Holonomy groups of Lorentzian manifolds

TL;DR

This paper surveys recent classification results of connected holonomy groups in Lorentzian manifolds, simplifies metric constructions, and explores applications including Einstein equations, spinor fields, and 2-symmetric manifolds.

Contribution

It provides a simplified construction method for Lorentzian metrics with all possible connected holonomy groups and discusses various applications.

Findings

Classification of connected holonomy groups is comprehensive.

Simplified construction of Lorentzian metrics with specific holonomy.

Applications to Einstein equations and special Lorentzian manifolds.

Abstract

In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian manifolds with parallel and recurrent spinor fields, conformally flat Walker metrics and the classification of 2-symmetric Lorentzian manifolds are considered.