Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem

Tongzhu Li; Xiang Ma; Changping Wang

doi:10.1016/j.aim.2014.02.002

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arXiv · 1204.1408

Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem

Tongzhu Li·Xiang Ma·Changping Wang

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Abstract

A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2. In this paper we consider Moebius rigidity for hypersurfaces and deformations of a hypersurface preserving the Moebius metric in the high dimensional case n>3. When the highest multiplicity of principal curvatures is less than n-2, the hypersurface is Moebius rigid. Deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a Reduction Theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future.

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BibTeX RIS

Tongzhu Li, Xiang Ma, Changping Wang. 2014-02-14. Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem. https://doi.org/10.1016/j.aim.2014.02.002

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