Curvature pinching for complete Kähler submanifolds of a complex projective space (Q1943251)

Let \(P_{n+p}(\mathbb C)\) be an \((n+p)\)-dimensional complex projective space endowed with the Fubini-Study metric of constant holomorphic sectional curvature \(1\) and \(M_{n}\) an \(n\)-dimensional complete Kähler submanifold immersed in \(P_{n+p}(\mathbb C)\). Let \(\rho\) denote the scalar curvature of \(M_{n}\). The author proves that if \(\rho >n^{2}\) then \(M_{n}\) is totally geodesic in \(P_{n+p}(\mathbb C)\).