Equivalent Killing vector fields via an isometry

Question

Suppose that $(M,g)$ is a complete semi-Riemannian manifold. We say that two Killing vector fields $V$ and $W$ are equivalent if there is $\Phi:M\rightarrow M$ an isometry such that $\Phi_*(V)=W$.

If we call $I(M)$ the isometry group of $M$ and $i(M)$ its Lie algebra (which is identified with Killing vectors fields on $M$), then the equivalent Killing vector fields to a given one $v\in i(M)$ are $Im f_v$, where $f_v:I(M)\rightarrow i(M)$ is given by $f_v(\Phi)=Ad_\Phi(v)$.

I am interesting on knowing when two timelike Killing vector field in the anti De Sitter space $\mathbb{H}^4_1$ are equivalent. I already know that in $\mathbb{H}^2_1$ any two timelike Killing vector fields are equivalent, but a similar result can not be true in $\mathbb{H}^3_1$ nor $\mathbb{H}^4_1$.

Anyone knows a reference about this?

Thanks in advance.

Answer 1

I believe that the answer to your question can be found in a paper I wrote with Joan Simón sa decade ago: Supersymmetric Kaluza-Klein reductions of AdS backgrounds (links to arXiv abstract); although you will probably need to do some work to extract the information.

In that paper we studied the one-parameter subgroups of isometries of anti de Sitter spacetimes in low dimension and in so doing classified the orbits in the Lie algebra of isometries under the adjoint action of the isometry group. The results for 4-dimensional AdS are contained in Section 4.2 and for 5-dimensional AdS in Section 4.3. Towards the end of the introduction of the paper there are some remarks about the notation which, in our own words, should it make it

possible to use our results without the time-consuming — albeit ultimately rewarding — task of reading the rest of the paper :)

Having said that, perhaps the following is helpful. The answer to your question for AdS${}_4$ is contained in pages 15 and 16. In the bottom half of page 15 you will find a list of the 15 orbits and in the list in page 16 the causal character of the corresponding Killing vectors. The causal character varies in many cases.

For AdS${}_5$, the similar lists are in pages 17 (bottom of the page) through 19.