Immersion of Kähler manifolds in the class of convex submanifolds
type: Item
instance of: scholarly article
title: Immersion of Kähler manifolds in the class of convex submanifolds
zbMATH Open document ID: 1372.53074
DOI: 10.1134/S0001434616090236
author: Alexander A. Borisenko
published in: Mathematical Notes
publication date: 2017-02-09T00:00:00Z
review text: 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-surface…
reviewed by: Maria Falcitelli
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 6682266
zbMATH Keywords: conex submanifold
zbMATH Keywords: Kähler manifold
zbMATH Keywords: null-index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6682266