Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry ${\text{Sol}}_0^4$

We study hypersurfaces of the four-dimensional Thurston geometry ${\text{Sol}}_0^4$, 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 hypersurfaces with parallel second fundamental form, and of totally umbilical hypersurfaces of ${\text{Sol}}_0^4$. We also give a closed expression for the Riemann curvature tensor of ${\text{Sol}}_0^4$, using two integrable complex structures.