a1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh-11451, Saudi Arabia
Simons  has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki  improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfies
then either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen  but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791).
(Received June 23 1995)