On a class of contact Riemannian 3-manifolds
Let \(M^3\) be a complete contact metric manifold and \((\omega, \xi, g, \varphi)\) its contact metric structure where \(\omega\) is a contact 1- form, \(\xi\) the characteristic vector field, \(g\) an associated Riemannian metric and \(\varphi\) a tensor field of type \((1,1)\). The author proves that the following four conditions are equivalent: 1) \(\xi\) is an eigenvector of the Ricci operator \(Q\) with constant eigenvalue; 2) \(Q= aI+ b\omega \otimes \xi+ ch\), \(a,b,c= \text{const.}\), where \(2h\) is the Lie differentiation of \(\varphi\) with respect to \(\xi\); 3) \(Q\varphi- \varphi Q= dh\varphi\), \(d= \text{const.}\); 4) \(\xi\) belongs to the \((k,\mu)\)-nullity distribution. Moreover, using a result of the paper [\textit{D. E. Blair}, \textit{T. Koufogiorgos} and \textit{B. J. Papantoniou}, Isr. J. Math. 91, No. 1-3, 189-214 (1995)]\ where condition 4 was studied, the author obtains that each of the above conditions implies that \(M^3\) is either a Sasakian manifold or locally isometric to one of the following Lie groups with a left invariant metric: SU(2) (or SO(3)), \(\text{SL} (2, \mathbb{R})\) (or O(1,2)), E(2) (the group of rigid motions of the Euclidean 2-space), E(1,1) (the group of rigid motions of the Minkowski 2-space).
scientific article; zbMATH DE number 785196
scientific article