Preudoumbilical Submanifolds In A Riemann Manifold Of Quasi Constant Curvature

This paper deals with submanifolds of the space N^n+p of quasi constant curvature. We discuss the submanifold with parallel mean curvature vector. Obtain a few pinching theorems about preudoumbilical submanifolds. At last the complete hypersurface with constant mean curvature in the quasi constant curvature space is investigated, some characterization of totally umbilical hypersurfaces are obtained. As the following :1. Let Mⁿ be a compact submanifold with parellel mean carvature in a Riemann manifold N^n+p of quasi constant curvature. If the sectional curvature R_ijij> 0, and η∈TM ,then Mⁿ is a predoumbilical submanifold.2. Let Mⁿ be a compact mininal submanifold with parallel the second fundmental form in a Riemann manifold N^n+p of quasi constant curvature. If b ≤ 0, then S ≤p(na+bi|∑η_i²)3. Let Mⁿ be a compact preudoumbilical submanifold with parallel mean curvature in a locally symmetric Riemann manifold N^n+p of quasi constant curvature. If the Mⁿ is a totally umbilical submanifold.4. Let Mⁿ be a compact hypersurface with constant mean curvature in a Riemann manifold Nⁿ⁺¹ of quasi constant curvature. If η∈TM, and a-2|b| = const > 0, then Mⁿ is a totally umbilical hypersurface when...