On minimal and Kähler submanifolds with isotropic second fundamental form (Q914170)

An isometrically immersed submanifold is said to have isotropic second fundamental form if the length of the second fundamental form related to any normal vector is the same one. In this letter, the author gives some curvature pinching theorems for compact minimal (resp. Kaehler) submanifolds in \(S^{n+p}(c)\) (resp. \({\mathbb{C}}P^{n+p}(1))\) with isotropic second fundamental form. In the case of real minimal submanifolds in \(S^{n+p}(c)\) the pinching condition characterizes the compact symmetric spaces of rank 1. In the case of Kaehler submanifolds in \({\mathbb{C}}P^{n+p}(1)\) a characterization of seven compact Kaehler submanifolds with parallel second fundamental form by the Ricci curvature pinching is shown.