Minimal submanifolds of a sphere with bounded second fundamental form

Let h be the second fundamental form of an n-dimensional minimal submanifold M of a unit sphere Sn+P (p > 2), S be the square of the length of h, and a(u) = IIh(u, u)112 for any unit vector u E TM. Simons proved that if S < n/(2 1/p) on M, then either S =_ 0, or S = n/(2 l/p). Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying S _ n/(2 l/p). In this paper the analogous results for a(u) are obtained. It is proved that if a(u) < 1, then either a(u) _ 0, or a(u) = 1 All minimal submanifolds satisfying a(u) are determined. A stronger result is obtained if M is odd-dimensional.