Three dimensional contact metric manifolds with vanishing Jacobi operator (Q839715)

In contact geometry, the Jacobi operator \(l=R(.,\xi )\xi \) plays a fundamental role. In the paper under review, the authors study 3-dimensional contact metric manifolds with \(l=R(.,\xi )\xi =0\). They denote by \(M_{l}\)-manifold a 3-dimensional contact metric manifold \(M(\eta,\xi,\phi,g)\) satisfying \(l=0\). They locally determine \(M_{l}\)-manifolds. For any function \(G:V\subseteq \mathbb R^{3}\rightarrow \mathbb R\) differentiable on an open subset \(V\subseteq \mathbb R^{3}\) such that \(\frac{\partial ^{2}G}{\partial x^{2}}=0\), a family of contact metric manifolds \(V(\eta,\xi,\phi,g)\) with \(l=0\) are constructed. They also classify \(M_{l}\)-manifolds with \(\left\| Q\xi \right\| =c=\)constant\(\geq 0\), where \(Q\) denotes the Ricci operator. Some nice examples are also given.