A criterion for the existence of Killing vectors in 3D

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or no independent Killing vectors. We present an explicit algorithm for the computing dimension of infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curvature, but they arc 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.