An extension of a result of H. Hopf to Kähler submanifolds of \(\mathbb{R}^ n\) (Q1915398)

A classical result of H. Hopf says that a constant mean curvature surface immersed in \(\mathbb{R}^3\) and homeomorphic to a sphere is a standard round sphere. The constancy of the mean curvature is equivalent to \(\nabla^\perp \alpha^{(1, 1)}= 0\), where \(\alpha^{(1, 1)}\) is the \((1, 1)\)-component of the complex bilinear extension of the second fundamental form of the surface to its complexified tangent bundle. This observation leads the authors to the following generalization of Hopf's result: Let \(M\) be a compact, connected, simply connected Kähler manifold with positive first Chern class and \(f: M\to \mathbb{R}^n\) an isometric immersion with \(\nabla^\perp \alpha^{(1, 1)}= 0\). Then \(M\) is isometric to a Riemannian product \(M_1\times\cdots \times M_r\) of Kähler manifolds and \(f\) splits into \(f= f_1\times \cdots \times f_r: M_1\times \cdots \times M_r\to \mathbb{R}^{n_1}\times \cdots \times \mathbb{R}^{n_r}= \mathbb{R}^n\), where each \(f_i: M_i\to \mathbb{R}^{n_i}\) is minimal in some sphere and second-order isotropic. The case \(n= 3\) and codimension of \(M\) equal to one is the result by Hopf.