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arXiv · 2303.14105

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

Marie D'haene·Jun-ichi Inoguchi·Joeri Van der Veken
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Abstract

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

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Marie D'haene, Jun-ichi Inoguchi, Joeri Van der Veken. 2023-03-24. Parallel and totally umbilical hypersurfaces of the four-dimensional Thurston geometry $\text{Sol}^4_0$. https://doi.org/10.1002/mana.202300372

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