Darboux-Framed BCV-Legendre Curve and Associated Geometry

In this study, the components of the Darboux frame associated with a Legendre curve lying on a surface in the BCV-Sasakianspace are derived for the first time. The geometric invariants of thecurve, namely its normal curvature, geodesic curvature, and geodesic torsion, are explicitly computed. A canal surface is then constructed aroundthe Darboux-framed BCV-Legendre curve, leading to the formulation ofDarboux-framed BCV-Legendre tubes. The intrinsic and extrinsic geometry of these tubular surfaces is investigated by calculating their Gaussianand mean curvatures. Moreover, the relationships between the parameter curves and geodesics, asymptotic curves, and curvature lines on theDarboux-framed BCV-Legendre tubes are analyzed in detail. All of theseresults are obtained for the first time in the context of BCV-Sasakian geometry, thus providing an original and significant contribution to thedifferential geometry of contact manifolds