Space-like Killing vector of Robertson-Walker metric? In the book "Kinetic theory in the expanding Universe" (J. Bernstein, 1988, Camb. Univ. Press), it was stated that 

"for nonstationary Robertson-Walker
  matrixes [sic] there is no spacelike
  Killing vector."

(page 6, footnote.)
But we know this is not true, since one can see that the generators of spatial translations/rotations are manifestly space-like. (For example, see R. Maartens and S.D. Maharaj, Class. Quantum Grav. 3 (1986) 1005-1011, equation (1.7)).
Was the author of the book wrong, or was it a slip of the hand, where "spacelike" should have been "timelike" (but that would amount to tautology anyway, since "nonstationary" is defined in terms of the absence of a time-like Killing vector).
 A: From the book by Kolb and Turner "The Early Universe", chapter 3.5:

In the strictest mathematical sense it is not possible for the Universe to be in thermal equilibrium, as the FRW cosmological model does not possess  a time-like Killinig vector. 

(Emphasis is mine). So it confirms the "slip of the hand" hypothesis.   
A: The de Sitter metric 
$$
ds^2~=~dt^2~-~e^{\sqrt{\Lambda/3}t}(dr^2~+~r^2d\Omega^2)
$$
has this time dependent factor.  This prevents a time-like Killing vector, for any vector formed by $\sqrt{g_{rr}^{-1}}\partial/\partial t$, or something similar, will not constant on a timelike spacetime vector.  The same holds for spacelike directions, for space and time are interchangeable in $g_{rr}$.  This means global conservation laws are not defined.  Energy conservation in general relativity is a funny thing, and is not something which can be established.  In the case of cosmologies, these metrics FLRW etc, are type O Petrov-Penrose-Pirani solutions, which have no isometries or Killing vectors.
