Submanifolds of the unit sphere (Q932427)

The author gives a pinching condition involving the square \(S\) of the length of the second fundamental form of a submanifold \(N^n\) of codimension \(p\geq 2\) in the unit sphere \(N^n\) is also a sphere. His main theorem is: Let \(N^n\) be an \(n\)-dimensionsional connected submanifold in the unit sphere \(S^{n+p}\) with parallel mean curvature vector field. If a) \(\sup_{x\in N^n} S< C(n,p)\), where \[ c(n,p)= {2n\over 1-{1\over p-1}+ \sqrt{{(n-2)^2\over n-1}+ ({3p-4\over p-1})^2}};\quad C(n,2)= 2\sqrt{n-1}, \] then \(N^n\) is a sphere with radius \(r= \sqrt{n/(n+S)}\). b) For \(p\geq 2\) if \(N^n\) is compact and \(S\leq C(n,p)\) then \(N^n\) is the totally umbilic sphere \(S^n(\sqrt{n/(n+S)}\) or an hypersphere of the totally geodesic \(S^{n+1}\) in \(S^{n+p}\).