On curvature pinching for minimal and Kaehler submanifolds with isotropic second fundamental form (Q1192395)

An isometrically immersed submanifold is said to have isotropic second fundamental form if the length of the second fundamental form, projected to any unit normal vector, is the same. The author proves some curvature pinching theorems for compact minimal (resp. Kähler) submanifolds of the sphere (resp. of the complex projective space) with isotropic second fundamental form. The author proves: Theorem. Let \(M^ n\) be a compact minimal immersed submanifold of \(S^{n+p}(c)\) with isotropic second fundamental form. If the Ricci curvature satisfies \[ \text{Ric}(M) \geq \{n-1-{1\over 2}(n+p+2)^{-1}p(n+2)\}c. \] Then \(M\) is either totally geodesic or a compact symmetric space of rank one. Theorem. Let \(M^ n\) be a compact Kähler submanifold in \(\mathbb{C} P^{n+p}(1)\) with isotropic second fundamental form. If \[ \text{Ric}(M) \geq n(n+p+1)/2(2p+n) \] then \(M\) is either totally geodesic or an imbedded submanifold congruent to the standard imbedding of one of the following submanifolds: \[ \mathbb{C} P^ n(1/2);\;Q^ n; \mathbb{C} P^ s(1) \times \mathbb{C} P^ s(1);\;U(s+2)/U(s) \times U(2),\;s\geq 3;\;SO(10)/U(5);\;E_ 6/\text{Spin}(10) \times T. \] Both theorems are achieved by proving that the submanifold has parallel second fundamental form. This is done by using ideas of \textit{S. Montiel}, \textit{A. Ros} and \textit{F. Urbano} [Math. Z. 191, 537-548 (1986; Zbl 0563.53046)] and \textit{A. Ros} [J. Math. Soc. Japan 36, 433-448 (1984; Zbl 0546.53035)]. The author also proves other pinching results related with sectional and scalar curvatures.