Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry Sol04

We study hypersurfaces of the four-dimensional Thurston geometry Sol(0)(4), which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor-including totally geodesic hypersurfaces and hyper surfaces with parallel second fundamental form-and of totally umbilical hypersurfaces of Sol(0)(4).We also give a closed expression for the Riemann curvature tensor of Sol(0)(4),using two integrable complex structures.