Umbilic Points and Real Hyperquadrics

There exist polynomial identities asociated to normal form, which yield an existence and uniqueness theorem. The space of normalized real hypersurfaces has a natural group action. Umbilic point is defined via normal form. A nondegenerate analytic real hypersurface is locally biholomorphic to a real hyperquadric if and only if every point of the real hypersurface is umbilic. 0. Introduction An analytic real hypersurface M is said to be in Chern-Moser normal form if M is defined by the following equation near the origin: v = 〈z, z〉+ ∑ min(s,t)≥2 Fst(z, z̄, u) where 〈z, z〉 ≡ zz1 + · · ·+ zze − zze+1 − · · · − zzn for a positive integer e in n2 ≤ e ≤ n, and Fst(μz, νz̄, u) = μ νFst(z, z̄, u) for all complex numbers μ, ν, and the functions F22, F23, F33 satisfy the condition ∆F22 = ∆ F23 = ∆ F33 = 0. Here the operator ∆ is defined as follows: ∆ ≡ D1D1 + · · ·+DeDe −De+1De+1 − · · · −DnDn, Dk = ∂ ∂zk , Dk = ∂ ∂zk , k = 1, · · · , n. Then we have an existence theorem of a biholomorphic normalizing mapping(cf. [CM], [Pa2]). 1