Reconcile The Riemann Curvature Hypersurfaces Rigid Unit Ball

In this paper,we search the proprty of a hypersurface in the unit sphere by the Riemannian Ricci curvature tensor.Let Mⁿbe a hypersurface in the unit sphere and Ricci curvature tensor be statified with R_ij,k - R_ik,j = 0,we make use of the property to get some new theoremes.Theorem 3.1 Let Riemannian manifold Mⁿ with the harmonic Riemannian curature tensor be isometricly immersed into a constant curvature space Nⁿ⁺¹(c)of dimmension n+1. If the mean curvature H=const .then M = S^P Theorem 3.2 Let Mⁿbe a compact Riemann manifold with nonnegative sectional curvature and harmonic curvature tensor .If Mⁿ can be immered into Sⁿ⁺¹ as a hypersurface ,then either S^k(a) × 5^n-k(b) (a² + b² = 1)or Sⁿ.Theorem 3.3 Let Mⁿ be a hypersurface in the unit sphereSⁿ⁺¹ (1)and Q a codazzi tensor. If nH² ≤ S ≤ nH² + δ²,|H|(2 + (n - 4)H²)δ - n(n - 2)²H² - n(n - 2)²H⁴ < 0.then Mⁿ = Sⁿ(1)...