Killing vectors in Robertson-Walker mertric

[paweld]

Let's consider the following metric:
$$g=dt^2 - a^2(t)\frac{dx^2+dy^2+dz^2}{1+\frac{x^2+y^2+z^2}{4}}$$
It can be also express in different coordinates as:
$$g=dt^2 - a^2(t)\left( \frac{dr^2}{1-r^2} + r^2(\sin^2(\theta) d\varphi^2+\theta^2) \right)$$

Of course it admits killing vetctors which are generators of rotations:
$$L_z=x\partial_y - y\partial_x, \ldots$$
Can anyone found different killing vector?
This metric describes spatialy homogenous universe so it should have
translation symmetry but it's not apparent in this coordinates.
Why it's hidden?
Thanks for any replies.

 

[George Jones]

Sorry for picking this up so late. Did you mean to omit $k$, i.e., did you mean to consider only the $k = 1$ case? If so, do you know do which Riemannian manifold models each spatial section, and can you guess what the spatial isometry group is?

It might be helpful to consider first the $k=0$ case.