The dimension of leaf closures of K-contact flows
type: Item
instance of: scholarly article
title: The dimension of leaf closures of K-contact flows
zbMATH Open document ID: 0827.53023
DOI: 10.1007/BF02108291
author: Philippe Rukimbira
published in: Annals of Global Analysis and Geometry
publication date: 1995-12-13T00:00:00Z
review text: Let \(M\) be a \((2n + 1)\)-dimensional manifold with a contact structure \((\xi, \alpha, g)\). The author considers special vector fields, mainly those which are infinitesimal isometries. They arise naturally in the setting of isometric flows where they describe the leaf closures. By considering the abelian Lie subalgebra of Killing vector fields containing the characteristic vector field \(\xi\) the author shows that a leaf closure of a \(K\)-contact \((2n + 1)\)-dimensional compact manifold has dimension at most \(n + 1\). It is known that the basic first cohomology of a \(K\)-contact flow on a compact manifold is isometric to its first de Rham cohomology. It is therefore expected that the de Rham cohomology will affect the topology of those flows. The author shows how the first Betti number of the manifold puts some restrictions on the maximum dimension of leaf closures. Using the first cohomological rank \(r_1 (M)\) of\(M\), that is the rank of the vector space of harmonic vector fields on \(M\), the author proves that a leaf closure of the \(K\)-contact flow on a compact \(K\)-contact \((2n + 1)\)-dimensional manifold \(M\) with \(K\)-contact metric \(g\) has dimension at most \(\inf \{n + 1, 2n + 1 - r_1 (M)\}\).
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37C85
zbMATH DE Number: 762854
zbMATH Keywords: \(K\)-contact manifold
zbMATH Keywords: leaf closure
zbMATH Keywords: cohomology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 762854