  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.
