On complete developable submanifolds in complex Euclidean spaces
type: Item
instance of: scholarly article
title: On complete developable submanifolds in complex Euclidean spaces
zbMATH Open document ID: 1027.53061
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: The authors study immersed, complete, complex submanifolds \(M^n\) in \({\mathbb C}^N\) which they call developable, which means that the image of such a submanifold's Gauss map \(\Gamma\colon M^n \to G_{\mathbb C}(n,N)\) into the complex Grassmannian has dimension \(r<n\). Recall that \(\Gamma(x)\) is the subspace of \({\mathbb C}^N\) parallel to the tangent space \(T_x(M^n)\). The level sets of \(\Gamma\) are \((n-r)\)-dimensional linear subvarieties of \({\mathbb C}^N\). If \(r=1\) they are all parallel to each other, and hence \(M^n\) is a cylinder. This is the complex analogue due to Abe of the classical Hartman-Nirenberg theorem. When \(r\geq 2\), there are non-cylinder examples found by Dajczer-Gromoll, Bourgain, Wu and Vitter. In this interesting paper, the authors first prove that when \(r=2\) and \(M^n\) is not a cylinder, then \(M^n\) is the total space of a holomorphic fiber bundle over a Riemann surface. The second main result concerns the general case when the rank restriction is removed. In the last part of paper, the authors raise some questions related to topological aspects of developable submanifolds and the holomorphic deformability of a developable submanifold into a cylinder.
reviewed by: Haizhong Li
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 1893858
zbMATH Keywords: developable submanifold
zbMATH Keywords: Gauss map
zbMATH Keywords: holomorphic fiber bundle
zbMATH Keywords: Riemann surface
zbMATH Keywords: holomorphic deformability
author: Hung-Hsi Wu
author: Fangyang Zheng
MaRDI profile type: Publication
DOI: 10.4310/CAG.2002.V10.N3.A6
description: scientific article; zbMATH DE number 1893858