HYPERSURFACES WITH CONSTANT QUASI-GAUSS-KRONECKER CURVATURE IN Mn+1(c)

Abstract

"Let Mn be an n-dimensional (n ≥ 3) complete smooth connected and oriented hypersurface in a real space form Mn+1(c) (c = 0, 1, −1) with constant quasiGauss-Kronecker curvature and two distinct principal curvatures. Denoting by H the mean curvature, |A| 2 the squared norm of the second fundamental form and Kq the quasi-Gauss-Kronecker curvature of Mn , we obtain some characterizations of S k (a)×Rn−k or S k (a)×S n−k ( √ 1 − a 2) or S k (a)×Hn−k (− √ 1 + a 2) in terms of H, |A| 2 and Kq, where 1 ≤ k ≤ n−1 and S k (a) is the k-dimensional sphere with radius a"