Inequalities relating sectional curvatures of a submanifold to the size of its second fundamental form and applications to pinching theorems for submanifolds

Abstract

The Gauss curvature equation is used to prove inequalities relating the sectional curvatures of a submanifold with the corresponding sectional curvature of the ambient manifold and the size of the second fundamental form. These inequalities are then used to show that if a manifold M ¯ \overline M is δ \delta -pinched for some δ > 1 4 \delta > \tfrac {1}{4} , then any submanifold M M of M ¯ \overline M that has small enough second fundamental form is δ M {\delta _M} -pinched for some δ M > 1 4 {\delta _M} > \tfrac {1}{4} . It then follows from the sphere theorem that the universal covering manifold of M M is a sphere. Some related results are also given.