Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem

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.