Odd dimensional tori and contact structure
type: Item
instance of: scholarly article
title: Odd dimensional tori and contact structure
zbMATH Open document ID: 0896.53030
author: Mitsuhiro Itoh
published in: Proceedings of the Japan Academy. Series A
publication date: 1998-09-24T00:00:00Z
review text: Let \(M\) be a metric contact Riemannian manifold with structure \((\phi,\xi,\eta,g)\). \(M\) is called a K-contact Riemannian manifold if the structure vector field \(\xi\) is a Killing field with respect to \(g\). In this paper, the author shows that if a compact manifold of dimension \(2n+1\) satisfies the cohomological condition \(\bigwedge^{2n+1} H^1(M,{\mathbb R}) = H^{2n+1}(M, {\mathbb R})\), then \(M\) cannot be a K-contact Riemannian manifold. And since the odd-dimensional torus \(T^{2n+1}\) satisfies the above cohomological condition, it cannot carry a K-contact structure. The author also shows that \(T^5\) cannot have a contact form \(\eta\) of the form \(\eta= \cos z dx + \sin z dy + f du + g dv\), where \(f,g\) satisfy either \(\partial f/\partial y = \partial g/\partial x = 0\) or \(\partial f/\partial x = \partial g/\partial y= 0\).
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 1057078
zbMATH Keywords: K-contact manifold
zbMATH Keywords: odd-dimensional torus
reviewed by: Andrew Bucki
MaRDI profile type: Publication
full work available at URL: pjaa.73.58
OpenAlex ID: W2014294011
cites work: Contact manifolds in Riemannian geometry
cites work: On harmonic tensors in compact Sasakian spaces
DOI: 10.3792/PJAA.73.58
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description: scientific article; zbMATH DE number 1057078