A three-dimensional homogeneous Lorentzian manifold is either symmetric or locally isometric to a Lie group equipped with a left-invariant Lorentzian metric . We completely classify surfaces with parallel second fundamental form in all non-symmetric homogeneous Lorentzian three-manifolds. Interesting differences arise with respect to the Riemannian case studied in [11, 12].
"PARALLEL SURFACES IN THREE-DIMENSIONAL LORENTZIAN LIE GROUPS." Taiwanese J. Math. 14 (1) 223 - 250, 2010. https://doi.org/10.11650/twjm/1500405737