Euclidean hypersurfaces with semi-parallel Moebius second fundamental form
Mateus Antas, Fernando Manfio
TL;DR
The paper addresses the problem of classifying umbilic-free Euclidean hypersurfaces $f:M^{n}\to\mathbb{R}^{n+1}$ with semi-parallel Möbius second fundamental form and three distinct Möbius principal curvatures. It develops the Möbius-invariant framework (Möbius metric $\langle\cdot,\cdot\rangle^{*}$, Möbius shape operator $B$, Blaschke tensor $\hat{\psi}$, Möbius form $\omega$) and leverages the closedness of $\omega$ to obtain a frame with diagonalizable Moebius invariants, showing at most three Möbius principal curvatures. The authors prove a complete local classification: when the Möbius principal curvature multiplicities are $(1,1,\ge2)$, the hypersurface is locally Möbius equivalent to either a cone over a Clifford torus or a rotational hypersurface over a hyperbolic cylinder; when all multiplicities are at least two, the hypersurface has parallel Möbius second fundamental form. This work completes the classification program initiated by Hu–Xie–Zhai and clarifies the roles of Möbius invariants and curvature conditions in the geometry of semi-parallel Möbius submanifolds, with implications for conformal submanifold geometry in Euclidean space.
Abstract
In this paper, we classify Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form and three distinct principal curvatures. This completes the classification of such hypersurfaces initiated by Hu, Xie and Zhai in the article: Submanifolds with semi-parallel Mobius second fundamental form in the unit sphere. J.Geom. Anal. 33 (2023), 44pp.