On slant curves in Sasakian 3-manifolds

Abstract

A classical theorem by Lancret says that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant. In this paper we study Lancret type problems for curves in Sasakian 3-manifolds.

Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry

“…Some results already known from [8] and [12] are reobtained with particular choices of the parameters; but to the best of our knowledge, until now there have been no general expressions for Legendre curves in Nil 3 nor for their curvature and torsion in S 3 .…”

“…They form a two parameters family (with parameters denoted as l and m) containing, among others, some remarkable 3-manifolds: R 3 , S 3 , S 2 × R, H 2 × R and the 3-dimensional Heisenberg group Nil 3 . Recently, several studies have been devoted to special submanifolds in these spaces: parallel surfaces [2], biharmonic curves [6], [12] and [13], constant angle surfaces [15], graphs of constant mean curvature [19], biharmonic surfaces [21], higher order parallel and totally umbilical surfaces [23].…”

“…1) the rôle of Legendre curves in almost contact geometry is remarkable and wellknown; in [4] the reader finds an excellent survey on these curves, 2) although the literature on Legendre curves is rich ( [3], [5], [7], [16], [20], [24], [25]), slant curves have been studied until now only for the Sasakian geometry in [12], for the contact pseudo-Hermitian geometry in [14], for the f -Kenmotsu geometry in [10], in normal almost contact geometry in [8] and for warped products in [9]. Although some Bianchi-Cartan-Vranceanu metrics are almost contact metrics we prefer in the present work a unified treatment in order to emphasize the common properties of these metrics.…”

SLANT CURVES IN 3-DIMENSIONAL ALMOST f-KENMOTSU MANIFOLDS

“…Chen studied surfaces in hyperbolic 3-space H 3 with proper mean curvature vector fields in [9]. On the other hand, as the generalization of Legendre curve, the notion of slant curves was introduced in [10].…”

“…In our previous paper [10], we studied slant curves in Sasakian 3-manifolds. In [11], we have shown that biharmonic curves in Sasakian space forms are slant.…”