On real Kähler Euclidean submanifolds with non-negative Ricci curvature (Q1769378)

\textit{L. A. Florit} and \textit{F. Zheng} [Comment. Math. Helv. 74, No.1, 53--62 (1999; Zbl 0941.53014)] showed that, if \(f:M^{n}\rightarrow \mathbb R^{m+p}\), \(2p\leq m\), is an isometric immersion of an \(m\)-dimensional connected Riemannian manifold \(M^{m}\) with non-positive sectional curvature and negative Ricci curvature, into the Euclidean space \(\mathbb R^{m+p}\), then \(2p=m\) and \(f\) splits locally as a product of \(p\) surfaces in \(\mathbb R^{3}\). In this article, the authors study splitting results when the Riemannian manifold has a Kähler structure, and show: Let \(f:M^{2n}\rightarrow \mathbb R^{2n+p}\), \( p\leq n \), be an isometric immersion of a Kähler manifold with either non-negative Ricci curvature or non-negative\ holomorphic sectional curvature. Then the index of relative nullity \(\nu \) of \(f\) satisfies \(\nu \geq 2n-2p\). Moreover, if \(\nu =2n-2p\), then there is an open dense subset \( W\subset M^{2n}\) such that \(f\left| _{W}\right. \) splits locally as a product of \(p\) nowhere flat real Kähler Euclidean hypersurfaces with non-negative Ricci curvature.