Kaya, Filiz ErtemSenyurt, Suleyman2024-11-072024-11-0720242073-8994https://doi.org/10.3390/sym16030323https://hdl.handle.net/11480/16012Willmore defined embedded surfaces on f:S -> E-3, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of E-3, inducing a Riemannian structure on f(S). The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature H-2 on f(S). Our aim is to observe the Gaussian and mean curvatures of curve-surface pairs using embedded surfaces in different curve-surface pairs and to define some developable operations on their curve-surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve-surface and derive the new characterizations. This curve-surface pair is called the osculating Darboux curve-surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve-surface pair. We also obtained the D-scroll of the curve-surface pair and involute D-scroll of the curve-surface pair with some differential geometric elements and found D(alpha,M)(s) and D*(alpha,M)(s)-scrolls of the curve-surface pair (alpha,M).eninfo:eu-repo/semantics/openAccesscurve-surface pairembeddedcurvaturesGaussian curvaturemean curvatureosculating Darboux frameArticle16310.3390/sym160303232-s2.0-85188957370Q2WOS:001193464500001N/A