Deformation of hypersurfaces preserving the Möbius metric and a reduction theorem

A hypersurface without umbilics in the (n+1)$(n+1)$-dimensional Euclidean space f:Mⁿ→Rⁿ⁺¹$f:M^n\to R^{n+1}$ 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$n?3$. 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$n?4$. When the highest multiplicity of principal curvatures is less than n−2$n-2$, 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.