The relative nullity of complex submanifolds and the Gauss map (Q2790275)

Let \(M\) be a complex complete submanifold of \(\mathbb{C}P^n\) equipped with the Fubini-Studi metric. The index of relative nullity is the minimum \( \mu(M) =\min_{p \in M} \ker\,(\alpha_p)\) of the dimension of the kernel of the second fundamental forms \( \alpha_p\). The theorem of Abe states that if \(\mu(M) >0\), then \(M\) is a totally geodesic submanifold. The authors give a new short and geometric proof of this theorem based on Jacobi fields. As an application, they give some sufficient conditions when a complex complete submanifold \(M^m \subset \mathbb{C}^n\) of the complex Euclidean space \(\mathbb{C}^n\) splits as an extrinsic product of a leaf of the relative nullity distribution \(\ker \,\alpha_p \subset T_pM\) and a complex complete submanifold, i.e., is a cylinder. One such sufficient condition is the existence of an open submanifold \(U \subset M\) such that for \(p \in U\), one has \(\dim \ker\,{\alpha_p} = \mu(M)\) and \(\mathrm{Ric}(X,X) < c ||X||^2\) for all vectors \(X \in T_pU,\, p \in U\) normal to \(\ker\,\alpha_p \) where \(c <0\) is a constant.