A CHARACTERIZATION OF QUADRIC CONSTANT GAUSS-KRONECKER CURVATURE HYPERSURFACES OF SPHERES

Let M subset of Sn+1 be a complete orientable hypersurface with constant Gauss-Kronecker curvature G. For any v is an element of Rn+2, let us define the following two real functions l(v), f(v) : M -> R on M by l(v) (x) = < x, v > and f(v) (x) = <nu(x), v > with nu : M -> Sn+1 a Gauss map of M. In this paper, we show that if n = 3, l(v) = lambda f(v) for some nonzero vector v is an element of R-5 and some real number lambda, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in the result we just mentioned. We also show that if n = 4, l(v) = lambda f(v) for some nonzero vector v is an element of R-6 and some real number lambda and (lambda(2) - 1)(2) + (G - 1)(2) not equal 0, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. Moreover, we will give an example of a complete hypersurface in S-5 with constant Gauss-Kronecker curvature that satisfies the condition l(v) = lambda f (v) for some non zero v, which is neither a totally umbilical hypersurface nor a cartesian product of Euclidean spheres.