On some types of slant curves in contact pseudo-Hermitian 3-manifolds (Q2880819)

Following the introduction, the authors review contact Riemannian geometry and the related contact pseudo-convex pseudo-Hermitian structure including a discussion of the ``Tanaka-Webster connection'' or ``pseudo-Hermitian connection''.NEWLINENEWLINEFor a curve \(\gamma\) in a contact Riemannian 3-manifold, one can define a Frenet frame field \(\{T, N, B\}\) along \(\gamma\) using the pseudo-Hermitian connection \(\widehat\nabla\). In this setting, we have the Frenet-Serret equations: \(\widehat\nabla_TT=\widehat\kappa N\), \(\widehat\nabla_TN=-\widehat\kappa T+\widehat\tau B\), \(\widehat\nabla_TB=-\widehat\tau N\), where \(\widehat\kappa=\|\widehat\nabla_TT\|\) is the ``pseudo-Hermitian curvature'' of \(\gamma\) and \(\widehat\tau\) its ``pseudo-Hermitian torsion''. The authors assume throughout that \(\widehat\kappa\neq 0\). The curve \(\gamma\) is said to be a ``slant curve'' if it makes a constant angle with the Reeb vector field of the contact form. A slant curve for which the angle is \(\pi/2\) is traditionally called a ``Legendre curve''. The ``pseudo-Hermitian mean curvature vector field'' of \(\gamma\) is defined by \(\widehat H=\widehat\nabla_TT=\widehat\kappa N\). Denoting the corresponding normal connection by \(\widehat\nabla^\perp\), one can speak of normal vector fields being ``pseudo-Hermitian parallel''.NEWLINENEWLINEThe authors first show that \(\gamma\) is a curve with pseudo-Hermitian parallel mean curvature vector field if and only if it is a pseudo-Hermitian circle (\(\widehat\kappa\) constant). Moreover, if in addition \(\gamma\) is slant, it is a Legendre circle.NEWLINENEWLINEThe curve \(\gamma\) is said to be a curve with ``pseudo-Hermitian proper mean curvature vector field'' if \(\widehat\Delta\widehat H=\lambda\widehat H\) where \(\lambda\) is a non-zero \(C^\infty\) function and \(\widehat\Delta\) is the Laplacian. The authors prove that such a curve has pseudo-Hermitian proper mean curvature vector field if and only if it is a pseudo-Hermitian circle with \(\lambda=\widehat\kappa^2\) or a pseudo-Hermitian helix with \(\lambda=\widehat\kappa^2+\widehat\tau^2\).NEWLINENEWLINEThe curve \(\gamma\) is said to be a curve with ``pseudo-Hermitian proper mean curvature vector field in the normal bundle'' if \(\widehat\Delta^\perp\widehat H=\lambda\widehat H\) where \(\widehat\Delta^\perp\) is the Laplacian in the normal bundle and \(\lambda\) a non-zero \(C^\infty\) function. The authors prove that \(\gamma\) is slant with pseudo-Hermitian proper mean curvature vector field in the normal bundle if and only if it is either a Legendre curve satisfying \(\lambda=-\widehat\kappa''/\widehat\kappa\) with \(\widehat\kappa(s)\neq as+b\) for some constants \(a\) and \(b\), or a pseudo-Hermitian slant helix satisfying \(\lambda=\widehat\tau^2\).NEWLINENEWLINEIn [Bull. Aust. Math. Soc. 81, No. 1, 156--164 (2010; Zbl 1185.53048)], \textit{J. E. Lee} defined curves of pseudo-Hermitian \(AW(k)\)-type, for \(k=1,2,3\). The authors of the present paper conclude by discussing slant curves of these types.