A criterion for the existence of Killing vectors in 3D

TL;DR

This paper introduces an explicit algorithm to determine the number of Killing vectors in a 3D Riemannian manifold by analyzing curvature invariants, providing a practical tool for symmetry classification.

Contribution

It presents a new algorithm that computes the dimension of the isometry algebra in 3D manifolds based on curvature invariants, improving upon existing criteria.

Findings

Algorithm effectively classifies the number of Killing vectors

Curvature invariants used are relative and not scalar polynomial Weyl invariants

Comparison with known criteria enhances understanding of symmetry conditions

Abstract

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curvature, but they are not scalar polynomial Weyl invariants. We compare our obstructions to the existence of Killing vectors with the known existence criteria due to Singer, Kerr and others.