An optimal rigidity theorem for complete submanifolds in a sphere

It is proved that if Mn is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in Sn p(1), and if SUPM S<α(n, H), where,α(n,H)=n n3/2(n-1)H2-n(n-2)/2(n-1)√n2H4 4(n-1)H2 ,then Mn must be the totally umbilical sphere Sn(1/√1 H2). An example to show that the pinching constantα (n, H) appears optimal is given.