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We study hypersurfaces of the four-dimensional Thurston geometry$\text{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Liegroup. In particular, we give a full classification of hypersurfaces whosesecond fundamental form is a Codazzi tensor, including totally geodesichypersurfaces and hypersurfaces with parallel second fundamental form, and oftotally umbilical hypersurfaces of $\text{Sol}^4_0$. We also give a closedexpression for the Riemann curvature tensor of $\text{Sol}^4_0$, using twointegrable complex structures.

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