A criterion for the existence of Killing vectors in 3D

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.

Dimension of the isometry group in three-dimensional Riemannian spaces

“…Their approach is supported by the capital theorems by Eisenhart [3] and Kerr [4], and they employ conditions that are expressed in terms of the eigenvalues and eigenvectors of the Ricci tensor. This kind of invariant approach has been revisited recently [5] by offering some algorithms for computing the dimension of the isometry group. Lately [6] we have presented an IDEAL approach to the transitive case: we have given the necessary and sufficient (Intrinsic, Deductive, Explicit and ALgorithmic) conditions for a three-dimensional Riemannian metric to admit a transitive group of isometries, and we have also distinguished the three different groups G 6 , the three different groups G 4 and the ten Bianchi-Behr types G 3 in transitive action.…”

“…We also present explicit Ricci concomitants that allow us to distinguish between the three cases of G 3 on O 2 and to discriminate when the group G 2 is abelian. Our tensorial approach avoids obtaining the Ricci eigenvectors and eigenvalues, which are necessary in the algorithms presented in the recent paper by Kruglikov and Tomoda [5].…”

Counting the number of Killing vectors in a 3D spacetime

“…In this paper, we classify the number of local isometry group for a Lorentzian manifold of dimension 3, by presenting the explicit forms of the second obstruction matrix R R R cls.d in all classes. This survey is essentially based on the procedure developed in [12], but differs from it in that: In Lorentzian signature, there appear null KVs and the Ricci tensor is not always diagonalisable. It is this aspect that prohibits the direct application of the previous analysis of Riemannian case [12] and requires the separate study, complicating attempts to pin it down discursively.…”

“…This survey is essentially based on the procedure developed in [12], but differs from it in that: In Lorentzian signature, there appear null KVs and the Ricci tensor is not always diagonalisable. It is this aspect that prohibits the direct application of the previous analysis of Riemannian case [12] and requires the separate study, complicating attempts to pin it down discursively. The strategy employed here is similar in spirit to the Erlangen programme, since the symmetry is classified in terms of differential invariants.…”

Differential invariants of Kundt waves

“…In addition, a sufficient number of spi has never been specified in the literature 1 . For instance, while principally known to resolve the count of Killing vectors for Riemannian metrics [9,25,7], the number and complexity of the involved spi is beyond a reasonable computational capacity [16].…”