Deformation of hypersurfaces preserving the Möbius metric and a reduction theorem

A hypersurface without umbilics in the (n+1)-dimensional Euclidean space f:Mn→Rn+1 is known to be determined by the Möbius metric g and the Möbius second fundamental form B up to a Möbius transformation when n⩾3. In this paper we consider Möbius rigidity for hypersurfaces and deformations of a hypersurface preserving the Möbius metric in the high dimensional case n⩾4. When the highest multiplicity of principal curvatures is less than n−2, the hypersurface is Möbius rigid. When the multiplicities of all principal curvatures are constant, 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.