A sphere theorem for submanifolds in a manifold with pinched positive curvature (Q1376014)

In generalization of a result of P.-F. Leung, a sphere theorem is proved for a compact, simple connected submanifold, isometrically immersed in a Riemannian manifold. A submanifold \(M\) with these properties is homeomorphic to an \(n\)-sphere, if \(|\sigma (v,v) |^2 <4/9 (\delta- 1/4)\) for every tangent vector \(\nu\) of \(M\), where \(\sigma\) denotes the second fundamental form of \(M\).