Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

Abstract

AbstractWe study hypersurfaces of the four‐dimensional Thurston geometry , 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 . We also give a closed expression for the Riemann curvature tensor of , using two integrable complex structures.