Submanifolds In Complex Space Forms In Real

This paper consists of two parts. In the first part, we consider the compact totally real pseudoumbilical submanifolds Mn that have nonzero parallel normalized mean curvature vector in complex space forms M(n+p)(c). As a result, (i)Let K(x) be the function assign to each point x of M the infinimum of the sectional curvatures of M at that ponit, If K(x) satisfies, then M is totally umbilical.(ii)LetQ(x) be the function assign to each point x of M the infinimum of the Ricci curvatures of M at that ponit, If Q(x) satisfies Q >1/4(n - 2)(c + 4H2) -1/4nC, then M is totally umbilical, (iii)Let a be the square length of the second fundermental form, if σ satisfies σ , then M is totally umbilical. (iv) if the scalar curvature p satisfies p > 1/12(3n2 - 5n)(c + 4H2) -1/6 c, then M is totally umbilical. where H is the constant mean curvature.In the second part, we investigate the compact submanifolds M with the parallel isoperimetric section in the real space forms Rm(c) and prove that if there exists a parallel isoperimetric section ξ on M, and the sectional curvature of M is always greater than zero, then M is contained in a hyper-sphere; and get that the Gauss curvature of the compact surfaces M with constant mean curvature in constant curvature space R4(c) is always greater than zero, then M is a totally geodesic surface or a sphere, where an isoperimetric ξ on M means a unit normal vector field defined globally on M with M1(ξ) = constant .