Some remarks on foliatedS 1 bundles

Given f : π1S → SO(2,1)0 and M = L+/f(π1S) as in Proposition 2, and any bounded neighbourhood U of 0 in H1(π1S,R2+1), there exists C such that, letting M_C = { (x,y,t) : -t^2 + x^2 + y^2 ≤ -C }/f(π1S), there exists a family of flat Lorentz metrics on M_C parametrized by U such that the spacetime M_C(u) corresponding to u ∈ U has holonomy ρ : π1M_C → ISO(2,1) such that qρ = f and ρ(γ)x = f(γ)x + tγ

“Proposition 3. Given f : π 1 S → SO (2 , 1) 0 and M = L + /f ( π 1 S ) as in proposition 2, and any bounded neighbourhood U of 0 in H 1 ( π 1 S, R 2+1 ) there exists C such that, letting M C = { ( x, y, t ) : -t 2 + x 2 + y 2 ≤ -C } /f ( π 1 S ) , there exists a family of flat Lorentz metrics on M C parametrized by U”