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

We study hypersurfaces of the four-dimensional Thurston geometry $$\mathrm{Sol}^4_0$$, 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 $$\mathrm{Sol}^4_0$$. We also give a closed expression for the Riemann curvature tensor of $$\mathrm{Sol}^4_0$$, using two integrable complex structures.