On 3-dimensional contact metric manifolds with \(\nabla_\xi\tau=0\)
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.
scientific article; zbMATH DE number 1200029