Slant curves in three-dimensional \(f\)-Kenmotsu manifolds (Q439276)

Among \((2n+1)\)-dimensional manifolds with almost contact structure, \((\varphi,\xi,\eta,g)\), \(f\)-Kenmotsu manifolds are distinguished by the following property of the Levi-Civita connection: \((\nabla_X\varphi)(Y)=f(g(\varphi X, Y)\xi-\varphi(X)\eta(Y))\), where \(f\) is a strictly positive smooth function and \(df\wedge \eta=0\) holds, see [\textit{V. Mangione}, Int. J. Math. Math. Sci. 2008, Article ID 798317, 7 p. (2008; Zbl 1147.53050)]. In Section 1, the authors review Kenmotsu geometry and Frenet curves. In Section 2 they study slant (particularly Legendre) curves in three-dimensional \(f\)-Kenmotsu manifolds. Slant curves are characterized through the scalar product between the normal at the curve and the Reeb vector field. In Sections 3--4, the authors pay special attention to the hyperbolic space \(H^3(-1)\) with its canonical Kenmotsu structure. The classification of slant curves in \(H^3(-1)\) is provided as well as the characterization of slant curves with proper mean curvature vector field.