Minimal submanifolds in a Riemannian manifold with pinched positive curvature (Q1904269)

The author proves that any \(n\)-dimensional minimal compact submanifold \(M\) of a Riemannian \((n+ p)\)-dimensional manifold \(N\) is totally geodesic provided that the sectional curvature of \(N\) is positive and \(\delta\)-pinched with \(\delta> (2n- 2)/ (5n- 2)\), the norm of the second fundamental form \(\sigma\) of \(M\) satisfies the inequality \(|\sigma|^2< {2\over 9} ((5n- 2)\delta- 2(n- 1))\) and the curvature tensor \(R\) of \(N\) preserves the tangent bundle \(TM\), i.e. that \(R(X, Y)Z \in TM\) whenever \(X\), \(Y\) and \(Z\in TM\).