Unit Sphere With A Parallel Hypersurface The Parablaschke Tensor

Let x : Mⁿâ†'Sⁿ⁺¹ be an immersed umbilic-free hypersurface in the (n+1)-dimensional unite sphere Sⁿ⁺¹.According to Wang Changping's M(?)bius geometric theory of submanifolds, Mⁿ is associated with a so-called M(?)bius metric g ,a M(?)bius second fundamental form B ,a Blaschke tensor A and a M(?)bius formΦwhich are invariants of M n M(?)bius transformation group of S n+1.A classical theorem of M(?)bius geometric states that M n( n≥3)is in fact characterized by g and B up to M(?)bius equivalence.Let D = A +λB,whereλis a constant.Obviously, D is a symmetric (0,2)-tensor and a M(?)bius invariant. D is called parablaschke tensor of x .In this paper,we give a complete classification of all immersed hypersurfaces in sphere with parallel parablaschke tensors.For this classification,two kinds of new examples are constructed.