Normal Extrinsically Flat Surfaces Along Curves on Spacelike Surfaces in de Sitter 3-Space

This paper investigates the extrinsic differential geometry of curves lying on a spacelike surface embedded in de Sitter 3-space. By employing the de Sitter Darboux frame along such a curve, we introduce a spacelike osculating Darboux vector field satisfying τg2(s)−κg2(s)>0. This vector field is then used to construct a normal extrinsically flat surface. Our main results provide a complete classification of the generic singularities of this constructed surface in terms of the fundamental invariants of the original curve, revealing a direct correspondence with the order of contact between the curve’s binormal vector and a tangent de Sitter 2-space.