Geometric symmetries on Lorentzian manifolds

TL;DR

This paper explores geometric symmetries in Lorentzian manifolds, focusing on Lie derivatives to classify and find solutions in general relativity, and discusses relationships and examples of these symmetries.

Contribution

It introduces new classification schemes for symmetries in Lorentzian manifolds and analyzes their interrelationships with illustrative examples.

Findings

Classification schemes for symmetries are developed.

Relationships between different symmetries are elucidated.

Examples demonstrate the application of symmetry classifications.

Abstract

Lie derivatives of various geometrical and physical quantities define symmetries and conformal symmetries in general relativity. Thus we obtain motions, collineations, conformal motions and conformal collineations. These symmetries are used not only to find new solutions of Einstein's field equations but to classify the spaces also. Different classification schemes are presented here. Relationships between these symmetries are discussed and illustrating examples are presented.