Certain results on \(K\)-contact and \((k, \mu )\)-contact manifolds
type: Item
instance of: scholarly article
title: Certain results on \(K\)-contact and \((k, \mu )\)-contact manifolds
zbMATH Open document ID: 1175.53060
author: Ramesh Sharma
published in: Journal of Geometry
publication date: 2009-06-08T00:00:00Z
review text: A Ricci soliton is a generalization of an Einstein metric defined on a Riemannian manifold \((M,g)\) by the condition \(L_Vg+2Ric+2\lambda g=0\), where \(V\) is a complete vector field on \(M\) and \(\lambda\) is a constant. According to the sign of \(\lambda\), the soliton is said to be shrinking, steady, or expanding, respectively. If the vector field \(V\) is the gradient of a potential function \(-f\), then \(g\) is called a gradient Ricci soliton. A Ricci soliton on a compact manifold is a gradient Ricci soliton. In dimension 2 and 3, a Ricci soliton on a compact manifold has constant curvature. According to a previous result, a compact Einstein K-contact manifold is Sasakian. In the present work, this is generalized by considering a complete K-contact metric as a Ricci soliton. It is proved that, if the metric of a complete K-contact manifold \((M,g)\) is a gradient soliton, then \(g\) is a shrinking soliton and \((M,g)\) is compact Einstein and Sasakian. For the nongradient case, the soliton vector field is a Jacobi vector field along the geodesics of the Reeb vector field. Other results show that among complete and simply connected K-contact manifolds only the unit sphere admits a non-Killing holomorphically planar conformal vector field and, if a \((k,\mu)\)-contact manifold admits a non-zero holomorphically planar conformal vector field, then it is either Sasakian or locally isometric to \(E^3\) of \(E^{n+1}\times S^n(4)\).
reviewed by: Zdeněk Dušek
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 5562593
zbMATH Keywords: \(K\)-contact manifold
zbMATH Keywords: \((k, \mu )\)-contact manifold
zbMATH Keywords: Sasakian manifold
zbMATH Keywords: Ricci soliton
zbMATH Keywords: holomorphically planar conformal vector field
MaRDI profile type: Publication
full work available at URL: s00022-008-2004-5
OpenAlex ID: W2089359567
DOI: 10.1007/S00022-008-2004-5
description: scientific article; zbMATH DE number 5562593