On the nullity index of isometric immersions of Kähler manifolds (Q1409670)

Let \(\varphi:M\to N\) be an isometric immersion of a Kähler manifold with complex dimension \(m\) into an ambient space \(N\). The map \(\varphi\) is said to be \textit{pluriminimal} if, for any holomorphic curve \(c\) immersed in \(M\), the curve \(\varphi\circ c\) is minimal. When \(N\) is flat, \textit{M. Dajczer} and \textit{L. Rodríguez} [Duke Math. J. 53, 211--220 (1986; Zbl 0599.53005)] showed that the pluriminimal immersions are exactly the minimal ones. Set \(\nu(x)\) (resp. \(\nu^{(1,1)}(x)\)) the index of relative nullity of the second fundamental form \(\alpha\) of \(\varphi\) (resp. of the \((1, 1)\) component of \(\alpha\)) at \(x\in M\). In case \(N=\mathbb{R}^{2n +p}\), \(n\geq 2\), M. Dajczer and L. Rodríguez classified the minimal immersions when \(\nu\) is greater than \(2n-4\) everywhere. In this article \(N\) is a conformally flat manifold having \(\nu(x_0)>0\) or \(\nu^{(1,1)}(x_0)>0\) at some point \(x_0\in M\). The authors show, under certain conditions, that \(m=1\), i.e. \(M\) is a Riemannian surface.