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

DOI10.1002/MANA.202300372arXiv2303.14105MaRDI QIDQ6509314

Marie D'haene, Jun-Ichi Inoguchi, Joeri Van der Veken

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

Mathematics Subject Classification ID

Differential geometry of homogeneous manifolds (53C30) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42)

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