On a conjecture of K. Ogiue (Q791178)

In this note we prove the following result. ''Let \(M^ n\) be a complete Kaehler submanifold of complex dimension \(n\geq 2\) in a complex projective space \({\mathbb{C}}P^{n+m}(1)\) of complex dimension \(n+m\) endowed with the Study-Fubini metric of constant holomorphic sectional curvature 1. Then, if the sectional curvature K of \(M^ n\) satisfies \(K>1/8\), \(M^ n\) is a totally geodesic submanifold of \({\mathbb{C}}P^{n+m}(1).''\) This solves completely and in the affirmative a conjecture by K. Ogiue. It is the best possible result in this direction, in the sense that for each n and c, \({\mathbb{C}}P^ n(c)\) can be imbedded isometrically as a Kaehler submanifold in \({\mathbb{C}}P^{n(n+3)/2}(2c)\) which is not totally geodesic; in particular, putting \(2c=1\), the sectional curvature of the submanifold satisfies \(K\geq 1/8\).