Compact minimal CR-submanifolds with the least nullity in a complex projective space (Q1355885)

The author studies the second variation of minimal CR-submanifolds of a complex projective \(m\)-space \(\mathbb{C} P^m\) and obtains the following Theorem. Let \(M\) be an \(n\)-dimensional compact minimal CR-submanifold of \(\mathbb{C} P^m\). (1) If \(n\) is even, then the nullity of \(M\) satisfies \(\text{nul}(M)\geq 2\left({n\over 2}+1\right)\left(m-{n\over 2}\right)\), with equality holding if and only if \(M\) is a totally geodesic Kähler submanifold. (2) If \(n\) is odd and equal to \(m\), then the nullity of \(M\) satisfies \(\text{nul}(M)\geq {{n(n+3)}\over 2}\), with equality holding if and only if \(M\) is a totally real totally geodesic submanifold. (3) If \(n\) is odd and not equal to \(m\), then the nullity of \(M\) satisfies \(\text{nul}(M)\geq n+1 +2\left({{n+1}\over 2}+1\right)\left(m-{{n+1}\over 2}\right)\), with equality holding if and only if \(M=\pi\left(S^1\left(\sqrt{1\over {n+1}} \right)\times S^n\left(\sqrt{n\over {n+1}} \right)\right)\subset \mathbb{C} P^{(n+1)/2}\) where \(\mathbb{C} P^{(n+1)/2}\) is imbedded in \(\mathbb{C} P^m\) as a totally geodesic Kähler submanifold and \(\pi:S^{1+(n+1)/2}\to \mathbb{C} P^{(n+1)/2}\) is the Hopf fibration.