Deformation of Hypersurfaces Preserving the Möbius Metric and a Reduction Theorem

Details Deformation of Hypersurfaces Preserving the Möbius Metric and a Reduction Theorem Deformation of Hypersurfaces Preserving the Möbius Metric and a Reduction Theorem

2012-04-06

arxiv.org/abs/1204.1408v1

Mathematics

Differential Geometry

50 pages

Scientific paper

A hypersurface without umbilics in the $(n+1)$-dimensional Euclidean space $f: M^n\rightarrow R^{n+1}$ is known to be determined by the M\"{o}bius metric $g$ and the M\"{o}bius second fundamental form $B$ up to a M\"{o}bius transformation when $n\geq 3$. In this paper we consider M\"{o}bius rigidity for hypersurfaces and deformations of a hypersurface preserving the M\"obius metric in the high dimensional case $n\geq 4$. When the highest multiplicity of principal curvatures is less than $n-2$, the hypersurface is M\"obius 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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