On the Moebius deformable hypersurfaces

TL;DR

This paper completes the classification of Moebius deformable hypersurfaces in Euclidean space for dimensions five and higher, addressing gaps in previous work and expanding understanding of their geometric properties.

Contribution

It extends the classification of Moebius deformable hypersurfaces to include a large class of previously omitted examples for dimensions n≥5.

Findings

Complete classification of Moebius deformable hypersurfaces for n≥5.

Identification of a large class of previously omitted examples.

Clarification of the geometric structure of these hypersurfaces.

Abstract

In the article [\emph{Deformations of hypersurfaces preserving the M\"obius metric and a reduction theorem}, Adv. Math. 256 (2014), 156--205], Li, Ma and Wang investigated the interesting class of Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $f\colon M^n\to \mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. The classification of Moebius deformable hypersurfaces of dimension $n\geq 4$ stated in the aforementioned article, however, misses a large class of examples. In this article we complete that classification for $n\geq 5$.