On 3-dimensional contact metric manifolds with \(\nabla_\xi\tau=0\)
type: Item
instance of: scholarly article
title: On 3-dimensional contact metric manifolds with \(\nabla_\xi\tau=0\)
zbMATH Open document ID: 0905.53024
DOI: 10.1007/BF01237607
author: Florence Gouli-Andreou
author: Philippos J. Xenos
published in: Journal of Geometry
publication date: 1999-02-04T00:00:00Z
review text: Let \((M,\eta, \xi,g, \phi)\) be a contact metric manifold and \(\tau= L_\xi g\) its torsion, where \(L_\xi g\) is the Lie derivative of the contact metric \(g\) with respect to the characteristic vector field \(\xi\). The condition (*) \(\nabla_\xi \tau=0\) was considered for the first time by \textit{S. S. Chen} and \textit{R. S. Hamilton} [Lect. Notes Math. 1111, 279-305 (1985; Zbl 0561.53039)] in their study of compact contact 3-manifolds. For a compact contact 3-manifold, the reviewer [\textit{D. Perrone}, Kodai Math. J. 13, 88-100 (1990; Zbl 0709.53034)] showed that (*) is the critical point condition for the integral of the scalar curvature considered as a functional on the set of all associated metrics. A study of the standard contact metric structure on the unit tangent sphere bundle of a Riemannian manifold satisfying (*) was given by the reviewer in [J. Geom. 49, 178-188 (1994; Zbl 0794.53030)]. In the geometry of contact metric 3-manifolds the condition (*) also arises when \(\xi\) is Anosov and other conditions are satisfied (see [\textit{D. E. Blair}, ``Special directions on contact metric manifolds of negative \(\xi\)-sectional curvature'', Ann. Fac. Sci. Toulouse, VI. Sér., Math. (to appear)]. In the present paper, the authors give the classifications of the contact metric 3-manifolds satisfying (*), which have curvature harmonic or \(\eta\)-parallel Ricci-tensor, or cyclic \(\eta\)-parallel Ricci tensor.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1200029
zbMATH Keywords: curvature harmonic Ricci tensor
zbMATH Keywords: \(\eta\)-parallel Ricci-tensor
zbMATH Keywords: cyclic \(\eta\)-parallel Ricci tensor
zbMATH Keywords: contact metric 3-manifolds
reviewed by: Domenico Perrone
MaRDI profile type: Publication
cites work: On a type of contact manifolds
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description: scientific article; zbMATH DE number 1200029