SUBMANIFOLDS OF A HIGHER DIMENSIONAL SPHERE

Let M be an m-dimensional manifold immersed in S~(m+k)(r).Then △X=μH-(m/r~2)X,where X is the position vector of M and H is a unit normal vector field which is orthogonalto X everywhere.If M is a compact connected manifold with parallel mean curvature vector field ξimmersed inS~(m+k)(r),and the sectional curvature of M is not less than (1/2)((1/r~2)+|ξ|~2),thenM is a small sphere.For a compact connected hypersurface M in S~(m+1)(r),if the sectional curvature is non-nesative and the scalar curvature is proportional to the mean curvature everywhere,then M isa totally umbilical hypersurface or the multiplication of two totally umbilical submanifolds.

SUBMANIFOLDS OF A HIGHER DIMENSIONAL SPHERE

Let M be an m-dimensional manifold immersed in $\{S^{m + k}}(r)\]$. Then $\\Delta X = \mu H - \frac{m}{{{r^2}}}X\]$. where X is the position vector of M and H is a unit normal vector field which is orthogonal to X everywhere. If M is a compact connected manifold with parallel mean curvature vector field $\\xi \]$ immersed in $\{S^{m + k}}(r)\]$, and the sectional curvature of M is not less than $\\frac{1}{2}(\frac{1}{{{r^2}}} + {\left| \xi \right|^2})\]$, then M is a small sphere. For a compact connected hypersurface M in $\{S^{m + 1}}(r)\]$, if the sectional curvature is nonnegative and the scalar curvature is proportional to the mean curvature everywhere, then M is a totally umbilical hypersurface or the multiplication of two totally umbilical submanifolds.