Infinitesimally Moebius bendable hypersurfaces

2023/07/30 by Jimenez, M. I., Tojeiro, R.
#53B25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.16341

Abstract

Li, Ma and Wang have provided in [Deformations of hypersurfaces preserving the Möbius metric and a reduction theorem, Adv. Math. 256 (2014), 156--205] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces f\colon Mⁿ→ ℝⁿ⁺¹ that admit non-trivial deformations preserving the Moebius metric. For n≥ 5, the classification was completed by the authors in \citeJT2. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion f\colon Mⁿ→ ℝ^m into Euclidean space as a one-parameter family of immersions f_t\colon Mⁿ→ ℝ^m, with t∈ (-ε, ε) and f₀=f, such that the Moebius metrics determined by f_t coincide up to the first order. Then we characterize isometric immersions f\colon Mⁿ→ ℝ^m of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the umbilic-free Euclidean hypersurfaces of dimension n≥ 5 that admit non-trivial infinitesimal Moebius variations.

Related

• On the Moebius deformable hypersurfaces
• Normally flat submanifolds with semi-parallel Moebius second fundamental form
• The infinitesimally bendable Euclidean hypersurfaces
• The infinitesimal deformations of hypersurfaces that preserve the Gauss map
• Deformations of surfaces preserving conformal or similarity invariants