On the construction of contact submanifolds with prescribed topology.
type: Item
instance of: scholarly article
title: On the construction of contact submanifolds with prescribed topology.
zbMATH Open document ID: 1034.53088
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: Let \(C\) be a compact contact manifold of dimension \(2n+1\) with a completely nonintegrable hyperplane distribution \(D\) on \(C\). Locally, \(D\) is defined by the kernel of a 1-form \(\theta\), that is \(D_x= \text{Ker}(\theta(x))\), and the locally defined 2-form \(d\theta\) is nondegenerate when restricted to \(D\). A contact submanifold of the contact manifold \((C,D_C)\) is a triple \((N,D_N, i)\), where \((N,D_N)\) is a contact manifold and \(i: N\to C\) is an embedding such that \(D_N= i^{-1}_*(D_C)\). The authors prove the following theorem: Let \(E\to C\) be a rank \(r\) complex vector bundle over \(C\) \((r\leq n)\) with top Chern class \(c_r(E)\). Then, there exists a contact submanifold \(W\) of \(C\) realizing the Poincaré dual of \(C_r(E)\) on \(H_{2n+1-2r}(C,\mathbb{Z})\). This result is analogue in the contact category to \textit{S. K. Donaldson's} construction of symplectic submanifolds [J. Differ. Geom. 44, 666--705 (1996; Zbl 0883.53032)]. The main tool in the proof is to show the existence of sequences of sections which are asymptotically holomorphic in an appropriate sense, and follows in some part the proof of \textit{D. Auroux} [Geom. Funct. Anal. 7, 971--995 (1997; Zbl 0912.53020)].
Mathematics Subject Classification ID: 53D35
zbMATH DE Number: 1782668
zbMATH Keywords: contact submanifolds of contact manifolds
Wikidata QID: Q115202953
author: A. Ibort
author: David Martínez Torres
author: Francisco Presas
reviewed by: Akihiko Morimoto
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090347644
OpenAlex ID: W2113015983
DOI: 10.4310/JDG/1090347644
description: scientific article; zbMATH DE number 1782668