A GENERAL RIGIDITY THEOREM FOR COMPLETE SUBMANIFOLDS

Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature nor mal in a complete and simply connected Riemannian (n+p)-manifold Nn+p with positive sectional curvature. For given integers n, p and for a n onnegative constant H we find a positive number tau(n, p) is an elemen t of (0, 1) with the property that if the sectional curvature of N is pinched in [tau(n, p), 1], and if the squared norm of the second funda mental form is in a certain interval, then Nn+p is isometric to the st andard unit (n+p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.