The geometrical properties of the Smarandache curves on 3-dimension pseudo-spheres generated by null curves

In this paper, we presented the geometrical properties of Smarandache curves on 3-pseudo-spheres. These curves were examined in the context of the lightcone, de Sitter space, and anti-de Sitter space. By leveraging the curvature relationships between the null curve and its corresponding Smarandache curves, we established necessary and sufficient conditions. Additionally, we illustrated our main results through two examples.