On the type decomposition of the second fundamental form of a Kähler submanifold (Q1915399)

Let \(\varphi: M\to N\) be an isometric immersion from a Kähler manifold \(M\) into a Riemannian manifold \(N\) with second fundamental form \(\alpha\). Put \(P(X, Y):= (\alpha(X, Y)+ \alpha(JX, JY))/2\) and note that \(P\) vanishes precisely if \(\varphi\) is a \((1, 1)\)-geodesic map, that is, the \((1, 1)\)-component of the complex bilinear extension of \(\alpha\) to \(TM\otimes \mathbb{C}\) vanishes. In the present paper, the authors study immersions \(\varphi\) for which \(P\) is totally umbilical.