Inequalities Relating Sectional Curvatures of a Submanifold to the Size of its Second Fundamental Form and Applications to Pinching Theorems for Submanifolds

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 $\overline M$ is $\delta$-pinched for some $\delta > \tfrac {1}{4}$, then any submanifold $M$ of $\overline M$ that has small enough second fundamental form is ${\delta _M}$-pinched for some ${\delta _M} > \tfrac {1}{4}$. It then follows from the sphere theorem that the universal covering manifold of $M$ is a sphere. Some related results are also given.