Immersion of Kähler manifolds in the class of convex submanifolds
A splitting theorem for real, complete, convex Kähler submanifolds of the Euclidean space is the main result of this paper. Given an isometric immersion \(f: M\to\mathbb{R}^n\), let \(r(x,\xi)\) be the rank of the second quadratic form \(\alpha(x,\xi)\) at any point \(x\in M\) with respect to the normal direction \(\xi\). The rank of the second quadratic form of \(M\) at \(x\) is \[ r(x)= \max\{r(x,\xi); \xi\in T^\perp_x M\}. \] One also sets \(j(x,\xi)= \min(k_1, k_2)\), if the diagonal form of \(\alpha(x,\xi)\) has \(k_1\) positive and \(k_2\) negative terms. The type of the point \(x\) is defined as \[ j(x)= \min\{j(x,\xi);\;\xi\in T^\perp_x M,\;r(x,\xi)= r(x)\}. \] If \(j(x)=0\), there exists a normal direction \(\xi\) such that \(\alpha(x,\xi)\) is nonnegative definite. The submanifold \(M\) is said to be convex if at each point \(x\in M\) the type \(j(x)\) vanishes. The extrinsic null-index \(\nu\) is the dimension of the null-space of the second quadratic form. So, at any \(x\in M\), one sets \[ \nu(x)= \dim\{v\in T_xM;\;\alpha(v, w)= 0,\;\text{for all }w\in T_xM\}. \] On any open set where \(\nu\) is constant, the null-spaces determine a foliation whose leaves are geodesic both in \(M\) and in the ambient space \(\mathbb{R}^n\). The author derives a lower bound for the extrinsic null-index of a convex isometric immersion of a Kähler manifold, stating the following results. Theorem A: Let \(f: M^{2n}\to\mathbb{R}^{2n+p}\) be an isometric immersion of a Kähler manifold. If \(M^{2n}\) is convex, one has \(\nu\geq 2(n-p)\). Moreover, if \(\nu= 2(n-p)\), then there exists an open dense subset \(W\) of \(M^{2n}\) such that \(f_{|W}\) splits, locally, as a product of Kähler hypersurfaces with nonnegative Ricci curvature. Theorem B: Let \(f: M^{2n}\to\mathbb{R}^{2n+p}\), \(p\leq n\), be an isometric immersion of a Kähler manifold. Assume that \(j(x)= 0\) and \(r(x)= 2n\) at each \(x\in M^{2n}\). Then \(p=n\) and \(f\) splits, locally, as a product of \(n\) 2-surfaces of positive Gaussian curvature in \(\mathbb{R}^3\). Moreover, if \(M^{2n}\) is complete, the splitting is global. Finally, the author extends a theorem stated by \textit{V. K. Beloshapka} and \textit{S. N. Bychkov} [Math. Notes 40, 854--857 (1986; Zbl 0618.32007); translation from Mat. Zametki 40, No. 5, 621--626 (1986)] for convex hypersurfaces in the space \(\mathbb C^n\). He proves that, if \(F^{2l+1}\) is a regular, convex submanifold of the complex Euclidean space \(\mathbb C^n\) and is foliated into complex planes \(\mathbb C^l\), then \(F^{2l+1}\) is a cylinder with \(\mathbb C^l\) as complex generator.
scientific article; zbMATH DE number 6682266