Geometric properties of Smarandache ruled surfaces generated by integral binormal curves in Euclidean 3-space

This paper investigates the geometric properties of Smarandache ruled surfaces generated by integral binormal curves in the Euclidean 3-space $E^3$. Specifically, we study four types of Smarandache ruled surfaces: the $tn$, $tb$, $nb$, and $tnb$ surfaces, each defined by different combinations of the tangent, normal, and binormal vectors of the integral curves. For each type of surface, we derive the parametric representations and compute the fundamental geometric properties, including the striction lines, distribution parameters, and the first and second fundamental forms. Additionally, we provide explicit expressions for the Gaussian and mean curvatures, which characterize the local shape of the surfaces. We also analyze the geodesic curvature, normal curvature, and geodesic torsion associated with the base curves on these surfaces. Furthermore, we establish necessary and sufficient conditions for these surfaces to be developable or minimal. The paper concludes with detailed conditions under which the base curves can be classified as geodesic or asymptotic lines on the surfaces. The results are supported by rigorous proofs and illustrative examples, offering a comprehensive understanding of the geometric behavior of these Smarandache ruled surfaces.