Hypersurfaces With Constant Mean Curvature In Locally Symmetric Space

In this paper orientable hypersurfaces with constant mean curvature(CMC) in locally symmetric space are concerned.Assumed that a compact CMC hypersurface Mⁿ is submerged in a 5 — Pinching ambient space Nⁿ⁺¹, and satisfieseverywhere in Mⁿ. ThenRijij ≥ 1 -δmeans that Mⁿ must be a totally umbilical submanifold of Nⁿ⁺¹, meanwhile Nⁿ⁺¹= Sⁿ⁺¹(1), or the sectional curvature R_ijij of Mⁿ varnish everywhere.In another result shows that if a compact CMC hypersurface satisfiesR_ijij (σ - nH²) > (1 -δ)σwhen λ_i is nonnegative everywhere where λ_i denote the principal curvatures of Mⁿ, then Nⁿ⁺¹ must bean dimensional unit spherical space, that is Nⁿ⁺¹ = Sⁿ⁺¹(1).In addition, a note about the relation of totally umbilical submanifolds and isotropic submanifolds is figured out. It says that a submanifold is totally umbilical if and only if it is a λ = (?)σ/n isotropic submanifold.